In this HackerRank Counting Road Networks problem solution, You must answer Q queries, where each query consists of some N denoting the number of cities Lukas wants to design a bidirectional network of roads for. For each query, find and print the number of ways he can build roads connecting n cities on a new line; as the number of ways can be quite large, print it modulo 663224321.
Problem solution in Java.
import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.InputMismatchException;
public class E2 {
InputStream is;
PrintWriter out;
String INPUT = "";
void solve()
{
int n = 100005;
long[] a = new long[n];
int mod = 663224321;
for(int i = 0;i < n;i++){
a[i] = pow(2, (long)i*(i-1)/2, mod);
}
int[][] fif = enumFIF(100005, mod);
long[] ta = transformLogarithmically(a, fif);
for(int Q = ni();Q > 0;Q--){
out.println(ta[ni()]);
}
}
public static int[][] enumFIF(int n, int mod) {
int[] f = new int[n + 1];
int[] invf = new int[n + 1];
f[0] = 1;
for (int i = 1; i <= n; i++) {
f[i] = (int) ((long) f[i - 1] * i % mod);
}
long a = f[n];
long b = mod;
long p = 1, q = 0;
while (b > 0) {
long c = a / b;
long d;
d = a;
a = b;
b = d % b;
d = p;
p = q;
q = d - c * q;
}
invf[n] = (int) (p < 0 ? p + mod : p);
for (int i = n - 1; i >= 0; i--) {
invf[i] = (int) ((long) invf[i + 1] * (i + 1) % mod);
}
return new int[][] { f, invf };
}
public static long pow(long a, long n, long mod) {
long ret = 1;
int x = 63 - Long.numberOfLeadingZeros(n);
for (; x >= 0; x--) {
ret = ret * ret % mod;
if (n << 63 - x < 0)
ret = ret * a % mod;
}
return ret;
}
public static int mod = 663224321;
public static int G = 3;
public static long[] mul(long[] a, long[] b)
{
return Arrays.copyOf(convoluteSimply(a, b, mod, G), a.length+b.length-1);
}
public static long[] mul(long[] a, long[] b, int lim)
{
return Arrays.copyOf(convoluteSimply(a, b, mod, G), lim);
}
public static long[] mulnaive(long[] a, long[] b)
{
long[] c = new long[a.length+b.length-1];
long big = 8L*mod*mod;
for(int i = 0;i < a.length;i++){
for(int j = 0;j < b.length;j++){
c[i+j] += a[i]*b[j];
if(c[i+j] >= big)c[i+j] -= big;
}
}
for(int i = 0;i < c.length;i++)c[i] %= mod;
return c;
}
public static long[] mulnaive(long[] a, long[] b, int lim)
{
long[] c = new long[lim];
long big = 8L*mod*mod;
for(int i = 0;i < a.length;i++){
for(int j = 0;j < b.length && i+j < lim;j++){
c[i+j] += a[i]*b[j];
if(c[i+j] >= big)c[i+j] -= big;
}
}
for(int i = 0;i < c.length;i++)c[i] %= mod;
return c;
}
public static long[] add(long[] a, long[] b)
{
long[] c = new long[Math.max(a.length, b.length)];
for(int i = 0;i < a.length;i++)c[i] += a[i];
for(int i = 0;i < b.length;i++)c[i] += b[i];
for(int i = 0;i < c.length;i++)if(c[i] >= mod)c[i] -= mod;
return c;
}
public static long[] add(long[] a, long[] b, int lim)
{
long[] c = new long[lim];
for(int i = 0;i < a.length && i < lim;i++)c[i] += a[i];
for(int i = 0;i < b.length && i < lim;i++)c[i] += b[i];
for(int i = 0;i < c.length;i++)if(c[i] >= mod)c[i] -= mod;
return c;
}
public static long[] sub(long[] a, long[] b)
{
long[] c = new long[Math.max(a.length, b.length)];
for(int i = 0;i < a.length;i++)c[i] += a[i];
for(int i = 0;i < b.length;i++)c[i] -= b[i];
for(int i = 0;i < c.length;i++)if(c[i] < 0)c[i] += mod;
return c;
}
public static long[] sub(long[] a, long[] b, int lim)
{
long[] c = new long[lim];
for(int i = 0;i < a.length && i < lim;i++)c[i] += a[i];
for(int i = 0;i < b.length && i < lim;i++)c[i] -= b[i];
for(int i = 0;i < c.length;i++)if(c[i] < 0)c[i] += mod;
return c;
}
public static long[] inv(long[] p)
{
int n = p.length;
long[] f = {invl(p[0], mod)};
for(int i = 0;i < p.length;i++){
if(p[i] == 0)continue;
p[i] = mod-p[i];
}
for(int i = 1;i < 2*n;i*=2){
long[] f2 = mul(f, f, Math.min(n, 2*i));
long[] f2p = mul(f2, Arrays.copyOf(p, i), Math.min(n, 2*i));
for(int j = 0;j < f.length;j++){
f2p[j] += 2L*f[j];
if(f2p[j] >= mod)f2p[j] -= mod;
if(f2p[j] >= mod)f2p[j] -= mod;
}
f = f2p;
}
for(int i = 0;i < p.length;i++){
if(p[i] == 0)continue;
p[i] = mod-p[i];
}
return f;
}
public static long[] d(long[] p)
{
long[] q = new long[p.length];
for(int i = 0;i < p.length-1;i++){
q[i] = p[i+1] * (i+1) % mod;
}
return q;
}
public static long[] i(long[] p)
{
long[] q = new long[p.length];
for(int i = 0;i < p.length-1;i++){
q[i+1] = p[i] * invl(i+1, mod) % mod;
}
return q;
}
public static long[] exp(long[] p)
{
int n = p.length;
long[] f = {p[0]};
for(int i = 1;i < 2*n;i*=2){
long[] ii = ln(f);
long[] sub = sub(ii, p, Math.min(n, 2*i));
if(--sub[0] < 0)sub[0] += mod;
for(int j = 0;j < 2*i && j < n;j++){
sub[j] = mod-sub[j];
if(sub[j] == mod)sub[j] = 0;
}
f = mul(sub, f, Math.min(n, 2*i));
}
return f;
}
public static long[] ln(long[] f)
{
long[] ret = i(mul(d(f), inv(f)));
ret[0] = f[0];
return ret;
}
public static long[] pow(long[] p, int K)
{
int n = p.length;
long[] lnp = ln(p);
for(int i = 1;i < lnp.length;i++)lnp[i] = lnp[i] * K % mod;
lnp[0] = pow(p[0], K, mod); // go well for some reason
return exp(Arrays.copyOf(lnp, n));
}
public static long[] divf(long[] a, int[][] fif)
{
for(int i = 0;i < a.length;i++)a[i] = a[i] * fif[1][i] % mod;
return a;
}
public static long[] mulf(long[] a, int[][] fif)
{
for(int i = 0;i < a.length;i++)a[i] = a[i] * fif[0][i] % mod;
return a;
}
public static long[] transformExponentially(long[] a, int[][] fif)
{
return mulf(exp(divf(Arrays.copyOf(a, a.length), fif)), fif);
}
public static long[] transformLogarithmically(long[] a, int[][] fif)
{
return mulf(Arrays.copyOf(ln(divf(Arrays.copyOf(a, a.length), fif)), a.length), fif);
}
public static long invl(long a, long mod) {
long b = mod;
long p = 1, q = 0;
while (b > 0) {
long c = a / b;
long d;
d = a;
a = b;
b = d % b;
d = p;
p = q;
q = d - c * q;
}
return p < 0 ? p + mod : p;
}
public static long[] reverse(long[] p)
{
long[] ret = new long[p.length];
for(int i = 0;i < p.length;i++){
ret[i] = p[p.length-1-i];
}
return ret;
}
public static long[] reverse(long[] p, int lim)
{
long[] ret = new long[lim];
for(int i = 0;i < lim && i < p.length;i++){
ret[i] = p[p.length-1-i];
}
return ret;
}
public static long[][] div(long[] p, long[] q)
{
if(p.length < q.length)return new long[][]{new long[0], Arrays.copyOf(p, p.length)};
long[] rp = reverse(p, p.length-q.length+1);
long[] rq = reverse(q, p.length-q.length+1);
long[] rd = mul(rp, inv(rq), p.length-q.length+1);
long[] d = reverse(rd, p.length-q.length+1);
long[] r = sub(p, mul(d, q, q.length-1), q.length-1);
return new long[][]{d, r};
}
public static long[] substitute(long[] p, long[] xs)
{
return descendProductTree(p, buildProductTree(xs));
}
public static long[][] buildProductTree(long[] xs)
{
int m = Integer.highestOneBit(xs.length)*4;
long[][] ms = new long[m][];
for(int i = 0;i < xs.length;i++){
ms[m/2+i] = new long[]{mod-xs[i], 1};
}
for(int i = m/2-1;i >= 1;i--){
if(ms[2*i] == null){
ms[i] = null;
}else if(ms[2*i+1] == null){
ms[i] = ms[2*i];
}else{
ms[i] = mul(ms[2*i], ms[2*i+1]);
}
}
return ms;
}
public static long[] descendProductTree(long[] p, long[][] pt)
{
long[] rets = new long[pt[1].length-1];
dfs(p, pt, 1, rets);
return rets;
}
private static void dfs(long[] p, long[][] pt, int cur, long[] rets)
{
if(pt[cur] == null)return;
if(cur >= pt.length/2){
rets[cur-pt.length/2] = p[0];
}else{
if(p.length >= 1500){
if(pt[2*cur+1] != null){
long[][] qr0 = div(p, pt[2*cur]);
dfs(qr0[1], pt, cur*2, rets);
long[][] qr1 = div(p, pt[2*cur+1]);
dfs(qr1[1], pt, cur*2+1, rets);
}else if(pt[2*cur] != null){
long[] nex = cur == 1 ? div(p, pt[2*cur])[1] : p;
dfs(nex, pt, cur*2, rets);
}
}else{
if(pt[2*cur+1] != null){
dfs(modnaive(p, pt[2*cur]), pt, cur*2, rets);
dfs(modnaive(p, pt[2*cur+1]), pt, cur*2+1, rets);
}else if(pt[2*cur] != null){
long[] nex = cur == 1 ? modnaive(p, pt[2*cur]) : p;
dfs(nex, pt, cur*2, rets);
}
}
}
}
public static long[][] divnaive(long[] a, long[] b)
{
int n = a.length, m = b.length;
if(n-m+1 <= 0)return new long[][]{new long[0], Arrays.copyOf(a, n)};
long[] r = Arrays.copyOf(a, n);
long[] q = new long[n-m+1];
long ib = invl(b[m-1], mod);
for(int i = n-1;i >= m-1;i--){
long x = ib * r[i] % mod;
q[i-(m-1)] = x;
for(int j = m-1;j >= 0;j--){
r[i+j-(m-1)] -= b[j]*x;
r[i+j-(m-1)] %= mod;
if(r[i+j-(m-1)] < 0)r[i+j-(m-1)] += mod;
}
}
return new long[][]{q, Arrays.copyOf(r, m-1)};
}
public static long[] modnaive(long[] a, long[] b)
{
int n = a.length, m = b.length;
if(n-m+1 <= 0)return a;
long[] r = Arrays.copyOf(a, n);
long ib = invl(b[m-1], mod);
for(int i = n-1;i >= m-1;i--){
long x = ib * r[i] % mod;
for(int j = m-1;j >= 0;j--){
r[i+j-(m-1)] -= b[j]*x;
r[i+j-(m-1)] %= mod;
if(r[i+j-(m-1)] < 0)r[i+j-(m-1)] += mod;
}
}
return Arrays.copyOf(r, m-1);
}
public static final int[] NTTPrimes = {1053818881, 1051721729, 1045430273, 1012924417, 1007681537, 1004535809, 998244353, 985661441, 976224257, 975175681};
public static final int[] NTTPrimitiveRoots = {7, 6, 3, 5, 3, 3, 3, 3, 3, 17};
public static long[] convoluteSimply(long[] a, long[] b, int P, int g)
{
int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
long[] fa = nttmb(a, m, false, P, g);
long[] fb = a == b ? fa : nttmb(b, m, false, P, g);
for(int i = 0;i < m;i++){
fa[i] = fa[i]*fb[i]%P;
}
return nttmb(fa, m, true, P, g);
}
public static long[] convolute(long[] a, long[] b)
{
int USE = 2;
int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
long[][] fs = new long[USE][];
for(int k = 0;k < USE;k++){
int P = NTTPrimes[k], g = NTTPrimitiveRoots[k];
long[] fa = nttmb(a, m, false, P, g);
long[] fb = a == b ? fa : nttmb(b, m, false, P, g);
for(int i = 0;i < m;i++){
fa[i] = fa[i]*fb[i]%P;
}
fs[k] = nttmb(fa, m, true, P, g);
}
int[] mods = Arrays.copyOf(NTTPrimes, USE);
long[] gammas = garnerPrepare(mods);
int[] buf = new int[USE];
for(int i = 0;i < fs[0].length;i++){
for(int j = 0;j < USE;j++)buf[j] = (int)fs[j][i];
long[] res = garnerBatch(buf, mods, gammas);
long ret = 0;
for(int j = res.length-1;j >= 0;j--)ret = ret * mods[j] + res[j];
fs[0][i] = ret;
}
return fs[0];
}
public static long[] convolute(long[] a, long[] b, int USE, int mod)
{
int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
long[][] fs = new long[USE][];
for(int k = 0;k < USE;k++){
int P = NTTPrimes[k], g = NTTPrimitiveRoots[k];
long[] fa = nttmb(a, m, false, P, g);
long[] fb = a == b ? fa : nttmb(b, m, false, P, g);
for(int i = 0;i < m;i++){
fa[i] = fa[i]*fb[i]%P;
}
fs[k] = nttmb(fa, m, true, P, g);
}
int[] mods = Arrays.copyOf(NTTPrimes, USE);
long[] gammas = garnerPrepare(mods);
int[] buf = new int[USE];
for(int i = 0;i < fs[0].length;i++){
for(int j = 0;j < USE;j++)buf[j] = (int)fs[j][i];
long[] res = garnerBatch(buf, mods, gammas);
long ret = 0;
for(int j = res.length-1;j >= 0;j--)ret = (ret * mods[j] + res[j]) % mod;
fs[0][i] = ret;
}
return fs[0];
}
private static long[] nttmb(long[] src, int n, boolean inverse, int P, int g)
{
long[] dst = Arrays.copyOf(src, n);
int h = Integer.numberOfTrailingZeros(n);
long K = Integer.highestOneBit(P)<<1;
int H = Long.numberOfTrailingZeros(K)*2;
long M = K*K/P;
int[] wws = new int[1<<h-1];
long dw = inverse ? pow(g, P-1-(P-1)/n, P) : pow(g, (P-1)/n, P);
long w = (1L<<32)%P;
for(int k = 0;k < 1<<h-1;k++){
wws[k] = (int)w;
w = modh(w*dw, M, H, P);
}
long J = invl(P, 1L<<32);
for(int i = 0;i < h;i++){
for(int j = 0;j < 1<<i;j++){
for(int k = 0, s = j<<h-i, t = s|1<<h-i-1;k < 1<<h-i-1;k++,s++,t++){
long u = (dst[s] - dst[t] + 2*P)*wws[k];
dst[s] += dst[t];
if(dst[s] >= 2*P)dst[s] -= 2*P;
// long Q = (u&(1L<<32)-1)*J&(1L<<32)-1;
long Q = (u<<32)*J>>>32;
dst[t] = (u>>>32)-(Q*P>>>32)+P;
}
}
if(i < h-1){
for(int k = 0;k < 1<<h-i-2;k++)wws[k] = wws[k*2];
}
}
for(int i = 0;i < n;i++){
if(dst[i] >= P)dst[i] -= P;
}
for(int i = 0;i < n;i++){
int rev = Integer.reverse(i)>>>-h;
if(i < rev){
long d = dst[i]; dst[i] = dst[rev]; dst[rev] = d;
}
}
if(inverse){
long in = invl(n, P);
for(int i = 0;i < n;i++)dst[i] = modh(dst[i]*in, M, H, P);
}
return dst;
}
static final long mask = (1L<<31)-1;
public static long modh(long a, long M, int h, int mod)
{
long r = a-((M*(a&mask)>>>31)+M*(a>>>31)>>>h-31)*mod;
return r < mod ? r : r-mod;
}
private static long[] garnerPrepare(int[] m)
{
int n = m.length;
assert n == m.length;
if(n == 0)return new long[0];
long[] gamma = new long[n];
for(int k = 1;k < n;k++){
long prod = 1;
for(int i = 0;i < k;i++){
prod = prod * m[i] % m[k];
}
gamma[k] = invl(prod, m[k]);
}
return gamma;
}
private static long[] garnerBatch(int[] u, int[] m, long[] gamma)
{
int n = u.length;
assert n == m.length;
long[] v = new long[n];
v[0] = u[0];
for(int k = 1;k < n;k++){
long temp = v[k-1];
for(int j = k-2;j >= 0;j--){
temp = (temp * m[j] + v[j]) % m[k];
}
v[k] = (u[k] - temp) * gamma[k] % m[k];
if(v[k] < 0)v[k] += m[k];
}
return v;
}
void run() throws Exception
{
is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
out = new PrintWriter(System.out);
long s = System.currentTimeMillis();
solve();
out.flush();
if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
}
public static void main(String[] args) throws Exception { new E2().run(); }
private byte[] inbuf = new byte[1024];
public int lenbuf = 0, ptrbuf = 0;
private int readByte()
{
if(lenbuf == -1)throw new InputMismatchException();
if(ptrbuf >= lenbuf){
ptrbuf = 0;
try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
if(lenbuf <= 0)return -1;
}
return inbuf[ptrbuf++];
}
private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
private double nd() { return Double.parseDouble(ns()); }
private char nc() { return (char)skip(); }
private String ns()
{
int b = skip();
StringBuilder sb = new StringBuilder();
while(!(isSpaceChar(b))){ // when nextLine, (isSpaceChar(b) && b != ' ')
sb.appendCodePoint(b);
b = readByte();
}
return sb.toString();
}
private char[] ns(int n)
{
char[] buf = new char[n];
int b = skip(), p = 0;
while(p < n && !(isSpaceChar(b))){
buf[p++] = (char)b;
b = readByte();
}
return n == p ? buf : Arrays.copyOf(buf, p);
}
private char[][] nm(int n, int m)
{
char[][] map = new char[n][];
for(int i = 0;i < n;i++)map[i] = ns(m);
return map;
}
private int[] na(int n)
{
int[] a = new int[n];
for(int i = 0;i < n;i++)a[i] = ni();
return a;
}
private int ni()
{
int num = 0, b;
boolean minus = false;
while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
if(b == '-'){
minus = true;
b = readByte();
}
while(true){
if(b >= '0' && b <= '9'){
num = num * 10 + (b - '0');
}else{
return minus ? -num : num;
}
b = readByte();
}
}
private long nl()
{
long num = 0;
int b;
boolean minus = false;
while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
if(b == '-'){
minus = true;
b = readByte();
}
while(true){
if(b >= '0' && b <= '9'){
num = num * 10 + (b - '0');
}else{
return minus ? -num : num;
}
b = readByte();
}
}
private static void tr(Object... o) { System.out.println(Arrays.deepToString(o)); }
}
Problem solution in C++.
#include <bits/stdc++.h>
using namespace std;
const int MAXN = 300005;
const int root = 1489;
const int root_1 = 296201594;
const int root_pw = (1<<19);
const long long int MOD = 663224321;
long long int fact[MAXN], invfact[MAXN], pow2[MAXN], all_graphs[MAXN], poly1[MAXN], poly2[MAXN];
long long int power(long long int a, int b)
{
if(!b)
return 1;
long long int ans = power(a,b/2);
ans = (ans*ans)%MOD;
if(b%2)
ans = (ans*a)%MOD;
return ans;
}
void fft (vector<long long int> &a, bool invert)
{
int n = (int) a.size();
for (int i = 1, j = 0; i < n; ++i)
{
int bit = n >> 1;
for (; j >= bit; bit>>=1)
j-=bit;
j+=bit;
if(i < j)
swap(a[i], a[j]);
}
for (int len = 2; len <= n; len<<=1)
{
long long int wlen = invert ? root_1 : root;
for (int i = len; i < root_pw; i<<=1)
wlen = (wlen*wlen)%MOD;
for (int i = 0; i < n; i+=len)
{
long long int w = 1;
for (int j = 0; j < len/2; ++j)
{
int u = a[i+j], v = (a[i+j+len/2]*w)%MOD;
a[i+j] = u+v < MOD ? u+v : u+v-MOD;
a[i+j+len/2] = u-v >= 0 ? u-v : u-v+MOD;
w = (w*wlen)%MOD;
}
}
}
if(invert)
{
int nrev = power(n, MOD-2);
for (int i=0; i<n; ++i)
a[i] = (a[i]*nrev)%MOD;
}
}
void multiply(vector <long long int> &a, vector <long long int> &b, vector <long long int> &c)
{
vector <long long int> ta(a.begin(), a.end()), tb(b.begin(), b.end()), tc;
int sz = 2*a.size();
ta.resize(sz);
tb.resize(sz);
fft(ta,false);
fft(tb,false);
for (int i = 0; i < sz; ++i)
{
ta[i] = (1ll*ta[i]*tb[i])%MOD;
}
fft(ta,true);
c = ta;
}
void dnc(int l, int r)
{
if(l+1 == r)
{
poly2[l] = (all_graphs[l]*invfact[l-1] + MOD - poly2[l])%MOD;
}
else
{
int m = (l + r)/2;
dnc(l,m);
dnc(m,r);
}
vector <long long int> p1, p2, ans;
int sz = r - l;
for (int i = sz; i < 2*sz; ++i)
{
p1.push_back(poly1[i]);
}
for (int i = l; i < r; ++i)
{
p2.push_back(poly2[i]);
}
multiply(p1,p2,ans);
for (int i = 0; i < ans.size(); ++i)
{
int pos = l + sz + i;
poly2[pos] = (poly2[pos] + ans[i])%MOD;
}
}
int main()
{
fact[0] = invfact[0] = pow2[0] = all_graphs[0] = 1;
for (int i = 1; i < MAXN; ++i)
{
fact[i] = (fact[i-1]*i)%MOD;
invfact[i] = power(fact[i], MOD-2);
pow2[i] = (pow2[i-1]*2)%MOD;
all_graphs[i] = (all_graphs[i-1]*pow2[i-1])%MOD;
poly1[i] = (all_graphs[i]*invfact[i])%MOD;
}
dnc(1,(1<<17)+1);
int t;
cin>>t;
while(t--)
{
int x;
cin>>x;
cout<<(poly2[x]*fact[x-1])%MOD<<"n";
}
return 0;
}
Problem solution in C.
#include<stdio.h>
#define MAX_N 150000
#define MODULE 663224321
static long long f[MAX_N];
static long long g[MAX_N];
static long long factorial_inverse[MAX_N];
static long long factorial[MAX_N];
static long long x1[MAX_N];
static long long x2[MAX_N];
static long long y[MAX_N];
long long power(long long x, long long n)
{
long long res = 1;
for( ; n ; n >>= 1, x = x * x % MODULE )
{
if( n & 1 )
{
res = ( res * x ) % MODULE;
}
}
return res;
}
void Run(long long *x, long long n, long long rev)
{
for( long long i = 1, j, k, t ; i < n ; ++i )
{
for( j = 0, t = i, k = n >> 1 ; k ; t >>= 1, k >>= 1 )
{
j = j << 1 | t & 1;
}
if( i < j )
{
long long tmp = x[i];
x[i] = x[j];
x[j] = tmp;
}
}
for( long long s = 2, ds = 1 ; s <= n ; ds = s, s <<= 1 )
{
long long wn = power(3, (MODULE - 1) / s);
if( rev < 0 )
{
wn = power(wn, MODULE - 2);
}
for( long long k = 0 ; k < n ; k += s )
{
long long w = 1, t;
for( long long i = k ; i < k + ds ; ++ i, w = w * wn % MODULE )
{
x[i+ds] = ( x[i] - ( t = w * x[i+ds] % MODULE ) + MODULE ) % MODULE;
x[i] = ( x[i] + t ) % MODULE;
}
}
}
if( rev < 0 )
{
long long invn = power(n, MODULE - 2);
for( long long i = 0 ; i < n ; ++i )
{
x[i] = x[i] * invn % MODULE;
}
}
}
void divide_conquer(long long left, long long right)
{
if( left == right )
{
return;
}
long long mid = ( left + right ) / 2;
divide_conquer(left, mid);
long long n1;
for( n1 = 1 ; n1 <= right - left ; n1 <<= 1 );
for( long long i = 0 ; i < n1 ; ++i )
{
x1[i] = ( i + left <= mid ) ? f[i+left] * factorial_inverse[i+left-1] % MODULE : 0;
x2[i] = (i + left <= right) ? factorial_inverse[i+1] * g[i+1] % MODULE : 0;
}
Run(x1, n1, 1);
Run(x2, n1, 1);
for( long long i = 0 ; i < n1 ; ++i )
{
y[i] = x1[i] * x2[i] % MODULE;
}
Run(y, n1, -1);
for( long long i = mid + 1 ; i <= right ; ++i )
{
f[i] = ( f[i] - ( factorial[i-1] * y[i-left-1] % MODULE ) + MODULE ) % MODULE;
}
divide_conquer(mid + 1, right);
}
void initialize()
{
factorial[0] = 1;
factorial_inverse[0] = 1;
for( long long i = 1 ; i < MAX_N ; ++i )
{
factorial[i] = ( factorial[i-1] * i ) % MODULE;
factorial_inverse[i] = power(factorial[i], MODULE - 2);
g[i] = power(2, (long long)i * (long long)(i - 1) / 2);
f[i] = g[i];
}
divide_conquer(1, 100000);
}
int main()
{
int q, n;
initialize();
scanf("%d", &q);
for( long long i = 0 ; i < q ; ++i )
{
scanf("%d", &n);
printf("%lldn", f[n]);
}
return 0;
}