Submit Info #40105

Problem Lang User Status Time Memory
$\sum_{i=0}^{\infty} r^i i^d$ java uwi AC 1351 ms 177.72 MiB

ケース詳細
Name Status Time Memory
0_00 AC 85 ms 12.47 MiB
0_01 AC 90 ms 12.47 MiB
0_02 AC 1351 ms 177.71 MiB
2_00 AC 93 ms 12.47 MiB
2_01 AC 93 ms 12.47 MiB
2_02 AC 101 ms 12.72 MiB
2_03 AC 185 ms 21.03 MiB
2_04 AC 1338 ms 177.72 MiB
2_05 AC 1325 ms 177.72 MiB
example_00 AC 92 ms 12.48 MiB

import java.io.ByteArrayInputStream; import java.io.IOException; import java.io.InputStream; import java.io.PrintWriter; import java.util.ArrayList; import java.util.Arrays; import java.util.InputMismatchException; import java.util.List; import java.util.function.LongFunction; public class Main { static InputStream is; static PrintWriter out; static String INPUT = ""; public static final int mod = 998244353; public static long[] makeDPowers(int n, int d) { boolean[] isp = new boolean[n+1]; Arrays.fill(isp, true); long[] id = new long[n+1]; Arrays.fill(id, 1L); for(int i = 2;i <= n;i++){ if(isp[i]){ long v = pow(i, d); for(int j = i;j <= n;j+=i){ for(int k = j;k % i == 0;k /= i) { id[j] = id[j] * v % mod; } isp[j] = false; } } } if(d != 0)id[0] = 0; return id; } // f(r)/(r-1)^{d+1} static void solve() { int r = ni(); int d = ni(); if(d == 0){ out.println(invl(mod + 1 - r, mod)); return; } // T(n, k) = Sum_{j=0..k} (-1)^j * (k-j)^n * binomial(n+1, j) // sum_k T(n,k)r^k = sum_{j=0..n} sum_{k=j..n} (-1)^j * (k-j)^n * binomial(n+1, j) // = sum_{j=0..n} (-1)^j * binomial(n+1, j) * sum_{k=j..n} (k-j)^n * r^k // = sum_{j=0..n} (-1)^j * binomial(n+1, j) * r^j * sum_{k=0..n-j} k^n * r^k long[] id = makeDPowers(d, d); int[][] fif = enumFIF(d+1, mod); long ef = 0; long S = 0; long rr = 1; long rd = pow(r, d-1); long ir = invl(r, mod); for(int j = d;j >= 0;j--){ S += id[d-j] * rr; S %= mod; ef = -ef + C(d+1, j, mod, fif) * S % mod * rd; ef %= mod; rr = rr * r % mod; rd = rd * ir % mod; } if(ef < 0)ef += mod; out.println(ef * r % mod * invl(pow(mod+1-r, d+1), mod) % mod); } public static long C(int n, int r, int mod, int[][] fif) { if (n < 0 || r < 0 || r > n) return 0; return (long) fif[0][n] * fif[1][r] % mod * fif[1][n - r] % mod; } 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 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 pow(long a, long n) { // a %= 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 void main(String[] args) throws Exception { long S = System.currentTimeMillis(); is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes()); out = new PrintWriter(System.out); solve(); out.flush(); long G = System.currentTimeMillis(); tr(G - S + "ms"); } private static boolean eof() { if (lenbuf == -1) return true; int lptr = ptrbuf; while (lptr < lenbuf) if (!isSpaceChar(inbuf[lptr++])) return false; try { is.mark(1000); while (true) { int b = is.read(); if (b == -1) { is.reset(); return true; } else if (!isSpaceChar(b)) { is.reset(); return false; } } } catch (IOException e) { return true; } } private static byte[] inbuf = new byte[1024]; public static int lenbuf = 0, ptrbuf = 0; private static 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 static boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); } private static int skip() { int b; while ((b = readByte()) != -1 && isSpaceChar(b)) ; return b; } private static double nd() { return Double.parseDouble(ns()); } private static char nc() { return (char) skip(); } private static 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 static 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 static 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 static int[] na(int n) { int[] a = new int[n]; for (int i = 0; i < n; i++) a[i] = ni(); return a; } private static 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 static 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(); } } public static <T> List<T> trans(long[] a, LongFunction<T> f) { List<T> b = new ArrayList<>(); for(long v : a)b.add(f.apply(v)); return b; } private static void tr(Object... o) { if (INPUT.length() != 0) System.out.println(Arrays.deepToString(o)); } }