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jiit.cc
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jiit.cc
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// https://www.codechef.com/OCT19A/problems/JIIT
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
const int M = 998244353, N = 2000;
ll binom[N+1][N+1];
ll pow2[N+1];
ll invpow2[N+1];
ll powq[2 * (N + 1)];
int aa[2][N+1][N+1];
ll modinv(ll a, ll b) {
ll t, q, x0 = 0, x1 = 1;
if (b == 1) return 1;
while (a > 1) {
q = a / b;
t = b; b = a % b; a = t;
t = x0; x0 = x1 - q * x0; x1 = t;
}
if (x1 < b) x1 += b;
return x1;
}
int pow(ll a, ll b, int m) {
ll r = 1, e = a;
while (b) {
if (b & 1) (r *= e) %= m;
(e *= e) %= m;
b >>= 1;
}
if (r < 0) r += m;
return r;
}
ll b(int k, int n, ll q, int i) {
ll r = 0;
for (int s = -n; s <= n; s += 2)
(r += powq[s + N] * ll(aa[i][k][(n - s) / 2])) %= M;
(r *= invpow2[n]) %= M;
return r;
}
ll f(ll n, ll m, ll q, ll z) {
ll r = 0;
for (ll k1 = 0; k1 <= n; k1++) {
ll x = z - k1 * m;
ll y = n - 2 * k1;
if (!y && m * k1 == z) {
for (ll k2 = 0; k2 <= m; k2++)
(r += b(k1, n, q, 0) * b(k2, m, q, n != m) % M * (binom[n][k1] * binom[m][k2] % M)) %= M;
continue;
}
if (!y || x % y) continue;
ll k2 = x / y;
if (k2 < 0 || k2 > m) continue;
(r += b(k1, n, q, 0) * b(k2, m, q, n != m) % M * (binom[n][k1] * binom[m][k2] % M)) %= M;
}
if (r < 0) r += M;
return r;
}
int main() {
ios::sync_with_stdio(0);
cin.tie(0);
pow2[0] = 1, invpow2[0] = 1;
for (int i = 1; i <= N; i++) {
pow2[i] = pow2[i - 1] * 2;
if (pow2[i] >= M) pow2[i] -= M;
}
for (int i = 1; i <= N; i++) invpow2[i] = modinv(pow2[i], M);
for (int i = 0; i <= N; i++) {
binom[i][0] = binom[i][i] = 1;
for (int j = 1; j < i; j++) {
binom[i][j] = binom[i-1][j-1] + binom[i-1][j];
if (binom[i][j] >= M) binom[i][j] -= M;
}
}
ll t, n, m, q, z;
cin >> t;
for (ll i = 0; i < t; i++) {
cin >> n >> m >> q >> z;
if (n % 2 == m % 2) {
int k = max(m, n);
for (int s = -k; s <= k; s += 2) powq[N + s] = pow(s, q, M);
} else {
for (int s = -n; s <= n; s += 2) powq[N + s] = pow(s, q, M);
for (int s = -m; s <= m; s += 2) powq[N + s] = pow(s, q, M);
}
ll mink1 = n + 1, mink2 = m + 1;
for (ll k1 = 0; k1 <= n; k1++) {
ll x = z - k1 * m;
ll y = n - 2 * k1;
if (!y && m * k1 == z) {
if (mink1 > k1) mink1 = k1;
mink2 = 0;
continue;
}
if (!y || x % y) continue;
ll k2 = x / y;
if (k2 < 0 || k2 > m) continue;
if (mink1 > k1) mink1 = k1;
if (mink2 > k2) mink2 = k2;
}
if (mink1 > n || mink2 > m) {
cout << "0\n";
continue;
}
if (n == m && mink1 > mink2) mink1 = mink2;
for (int k = mink1; k <= n; k++) aa[0][k][0] = 1;
for (int u = 1; u <= n; u++) {
if (binom[n][u]) aa[0][n][u] = u & 1 ? M - binom[n][u] : binom[n][u];
else aa[0][n][u] = 0;
}
for (int u = 1; u <= n; u++)
for (int k = n - 1; k >= mink1; k--) {
ll x = aa[0][k+1][u-1] + aa[0][k+1][u];
if (x >= M) x -= M;
x += aa[0][k][u-1];
if (x >= M) x -= M;
aa[0][k][u] = x;
}
if (n != m) {
for (int k = mink2; k <= m; k++) aa[1][k][0] = 1;
for (int u = 1; u <= m; u++) {
if (binom[m][u]) aa[1][m][u] = u & 1 ? M - binom[m][u] : binom[m][u];
else aa[1][m][u] = 0;
}
for (int u = 1; u <= m; u++)
for (int k = m - 1; k >= mink2; k--) {
ll x = aa[1][k+1][u-1] + aa[1][k+1][u];
if (x >= M) x -= M;
x += aa[1][k][u-1];
if (x >= M) x -= M;
aa[1][k][u] = x;
}
}
cout << f(n, m, q, z) << "\n";
}
}