import std.conv, std.functional, std.range, std.stdio, std.string; import std.algorithm, std.array, std.bigint, std.bitmanip, std.complex, std.container, std.math, std.mathspecial, std.numeric, std.regex, std.typecons; import core.bitop; class EOFException : Throwable { this() { super("EOF"); } } string[] tokens; string readToken() { for (; tokens.empty; ) { if (stdin.eof) { throw new EOFException; } tokens = readln.split; } auto token = tokens.front; tokens.popFront; return token; } int readInt() { return readToken.to!int; } long readLong() { return readToken.to!long; } real readReal() { return readToken.to!real; } bool chmin(T)(ref T t, in T f) { if (t > f) { t = f; return true; } else { return false; } } bool chmax(T)(ref T t, in T f) { if (t < f) { t = f; return true; } else { return false; } } int binarySearch(alias pred, T)(in T[] as) { int lo = -1, hi = cast(int)(as.length); for (; lo + 1 < hi; ) { const mid = (lo + hi) >> 1; (unaryFun!pred(as[mid]) ? hi : lo) = mid; } return hi; } int lowerBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a >= val)); } int upperBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a > val)); } struct ModInt(uint M_) { import std.conv : to; alias M = M_; uint x; this(ModInt a) { x = a.x; } this(uint x_) { x = x_ % M; } this(ulong x_) { x = x_ % M; } this(int x_) { x = ((x_ %= cast(int)(M)) < 0) ? (x_ + cast(int)(M)) : x_; } this(long x_) { x = cast(uint)(((x_ %= cast(long)(M)) < 0) ? (x_ + cast(long)(M)) : x_); } ref ModInt opAssign(T)(inout(T) a) if (is(T == uint) || is(T == ulong) || is(T == int) || is(T == long)) { return this = ModInt(a); } ref ModInt opOpAssign(string op, T)(T a) { static if (is(T == ModInt)) { static if (op == "+") { x = ((x += a.x) >= M) ? (x - M) : x; } else static if (op == "-") { x = ((x -= a.x) >= M) ? (x + M) : x; } else static if (op == "*") { x = cast(uint)((cast(ulong)(x) * a.x) % M); } else static if (op == "/") { this *= a.inv(); } else static assert(false); return this; } else static if (op == "^^") { if (a < 0) return this = inv()^^(-a); ModInt b = this, c = 1U; for (long e = a; e; e >>= 1) { if (e & 1) c *= b; b *= b; } return this = c; } else { return mixin("this " ~ op ~ "= ModInt(a)"); } } ModInt inv() const { uint a = M, b = x; int y = 0, z = 1; for (; b; ) { const q = a / b; const c = a - q * b; a = b; b = c; const w = y - cast(int)(q) * z; y = z; z = w; } assert(a == 1); return ModInt(y); } ModInt opUnary(string op)() const { static if (op == "+") { return this; } else static if (op == "-") { ModInt a; a.x = x ? (M - x) : 0U; return a; } else static assert(false); } ModInt opBinary(string op, T)(T a) const { return mixin("ModInt(this) " ~ op ~ "= a"); } ModInt opBinaryRight(string op, T)(T a) const { return mixin("ModInt(a) " ~ op ~ "= this"); } bool opCast(T: bool)() const { return (x != 0U); } string toString() const { return x.to!string; } } enum MO = 998244353; alias Mint = ModInt!MO; enum LIM = 410; Mint[] inv, fac, invFac; void prepare() { inv = new Mint[LIM]; fac = new Mint[LIM]; invFac = new Mint[LIM]; inv[1] = 1; foreach (i; 2 .. LIM) { inv[i] = -((Mint.M / i) * inv[cast(size_t)(Mint.M % i)]); } fac[0] = invFac[0] = 1; foreach (i; 1 .. LIM) { fac[i] = fac[i - 1] * i; invFac[i] = invFac[i - 1] * inv[i]; } } Mint binom(long n, long k) { if (n < 0) { if (k >= 0) { return (-1)^^(k & 1) * binom(-n + k - 1, k); } else if (n - k >= 0) { return (-1)^^((n - k) & 1) * binom(-k - 1, n - k); } else { return Mint(0); } } else { if (0 <= k && k <= n) { assert(n < LIM); return fac[cast(size_t)(n)] * invFac[cast(size_t)(k)] * invFac[cast(size_t)(n - k)]; } else { return Mint(0); } } } // det(x I + a) // O(n^3) T[] charPoly(T)(const(T[][]) a) { import std.algorithm.mutation : swap; const n = cast(int)(a.length); auto b = new T[][](n, n); foreach (i; 0 .. n) b[i][] = a[i][]; // upper Hessenberg foreach (j; 0 .. n - 2) { foreach (i; j + 1 .. n) { if (b[i][j]) { swap(b[j + 1], b[i]); foreach (ii; 0 .. n) swap(b[ii][j + 1], b[ii][i]); break; } } if (b[j + 1][j]) { const r = 1 / b[j + 1][j]; foreach (i; j + 2 .. n) { const s = r * b[i][j]; foreach (jj; j .. n) b[i][jj] -= s * b[j + 1][jj]; foreach (ii; 0 .. n) b[ii][j + 1] += s * b[ii][i]; } } } // fss[i] := det(x I_i + b[0..i][0..i]) auto fss = new T[][n + 1]; fss[0] = [T(1)]; foreach (i; 0 .. n) { fss[i + 1] = new T[i + 2]; foreach (k; 0 .. i + 1) fss[i + 1][k + 1] = fss[i][k]; foreach (k; 0 .. i + 1) fss[i + 1][k] += b[i][i] * fss[i][k]; T prod = 1; foreach_reverse (j; 0 .. i) { prod *= -b[j + 1][j]; const s = prod * b[j][i]; foreach (k; 0 .. j + 1) fss[i + 1][k] += s * fss[j][k]; } } return fss[n]; } Mint solve(Mint[] as, Mint[] cs, long N) { const d = cast(int)(cs.length) - 1; Mint[] mul(Mint[] fs, Mint[] gs) { auto hs = new Mint[d + d - 1]; foreach (i; 0 .. d) foreach (j; 0 .. d) { hs[i + j] += fs[i] * gs[j]; } foreach_reverse (i; d .. d + d - 1) { foreach (j; 1 .. d + 1) { hs[i - j] -= cs[j] * hs[i]; } } hs.length = d; return hs; } auto xs = new Mint[d]; auto ys = new Mint[d]; xs[1] = 1; ys[0] = 1; for (long e = N; e; e >>= 1) { if (e & 1) ys = mul(ys, xs); xs = mul(xs, xs); } Mint ans; foreach (i; 0 .. d) { ans += as[i] * ys[i]; } return ans; } void main() { prepare; try { for (; ; ) { const S = readToken; const K = readLong; const N = cast(int)(S.length) / 2; // pos, depth, state auto dp = new Mint[][][](2 * N + 1, N + 2, N + 2); dp[0][0][0] = 1; foreach (i; 0 .. 2 * N) { foreach (j; 0 .. N + 1) foreach (u; 0 .. N + 1) { dp[i + 1][j + 1][u] += dp[i][j][u]; if (j > 0) { dp[i + 1][j - 1][u + ((S[i] == '(') ? 1 : 0)] += dp[i][j][u]; } } } debug { writeln("dp[2 N][0] = ", dp[2 * N][0]); } // initial state, final state auto mat = new Mint[][](N + 1, N + 1); foreach (u; 0 .. N + 1) { mat[u][u] += ((2 * N) * (2 * N - 1) / 2 - (N - u)^^2 - u^^2); if (u < N) mat[u][u + 1] += (N - u)^^2; if (u > 0) mat[u][u - 1] += u^^2; } auto cs = charPoly(mat); cs.reverse; for (int i = 1; i <= N + 1; i += 2) { cs[i] *= -1; } debug { writeln("mat = ", mat); writeln("cs = ", cs); } // # of ops, state auto DP = new Mint[][](N + 1, N + 1); DP[0][0] = 1; foreach (k; 0 .. N) { foreach (u; 0 .. N + 1) foreach (v; 0 .. N + 1) { DP[k + 1][v] += DP[k][u] * mat[u][v]; } } auto gs = new Mint[N + 1]; foreach (u; 0 .. N + 1) { auto as = iota(N + 1).map!(k => DP[k][u]).array; gs[u] = solve(as, cs, K); gs[u] /= binom(N, u)^^2; } debug { writeln("gs = ", gs); } Mint ans; foreach (u; 0 .. N + 1) { ans += dp[2 * N][0][u] * gs[u]; } writeln(ans); } } catch (EOFException e) { } }