Problem:
Score : 600 points

Problem Statement

Let us define the oddness of a permutation p = {p_1,\ p_2,\ ...,\ p_n} of {1,\ 2,\ ...,\ n} as \sum_{i = 1}^n |i - p_i|.

Find the number of permutations of {1,\ 2,\ ...,\ n} of oddness k, modulo 10^9+7.

Constraints

All values in input are integers.

1 \leq n \leq 50

0 \leq k \leq n^2

Input

Input is given from Standard Input in the following format:

n k

Output

Print the number of permutations of {1,\ 2,\ ...,\ n} of oddness k, modulo 10^9+7.

Sample Input 1

3 2

Sample Output 1

2

There are six permutations of {1,\ 2,\ 3}. Among them, two have oddness of 2: {2,\ 1,\ 3} and {1,\ 3,\ 2}.

Sample Input 2

39 14

Sample Output 2

74764168

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#ifdef stderr_path
#define LOCAL
#endif
#ifdef LOCAL
#define _GLIBCXX_DEBUG
#else
#pragma GCC optimize("Ofast")
#endif
#include <algorithm>
#include <bitset>
#include <cassert>
#include <chrono>
#include <complex>
#include <cstring>
#include <deque>
#include <functional>
#include <iomanip>
#include <iostream>
#include <map>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <unordered_map>
#include <unordered_set>

// #define NDEBUG
#define debug_stream std::cerr
#define iostream_untie true
#define __precision__ 10

#define all(v) std::begin(v), std::end(v)
#define rall(v) std::rbegin(v), std::rend(v)
#define __odd(n) ((n)&1)
#define __even(n) (not __odd(n))
#define __popcount(n) __builtin_popcountll(n)
#define __clz32(n) __builtin_clz(n)
#define __clz64(n) __builtin_clzll(n)
#define __ctz32(n) __builtin_ctz(n)
#define __ctz64(n) __builtin_ctzll(n)

using i64 = int_fast64_t;
using pii = std::pair<int, int>;
using pll = std::pair<int_fast64_t, int_fast64_t>;
template <class T>
using heap = std::priority_queue<T>;
template <class T>
using minheap = std::priority_queue<T, std::vector<T>, std::greater<T>>;
template <class T>
constexpr T inf = std::numeric_limits<T>::max() / T(2) - T(1123456);

namespace execution
{
    std::chrono::system_clock::time_point start_time, end_time;
    void print_elapsed_time()
    {
        end_time = std::chrono::system_clock::now();
        std::cerr << "\n----- Exec time : ";
        std::cerr << std::chrono::duration_cast<std::chrono::milliseconds>(
                         end_time - start_time)
                         .count();
        std::cerr << " ms -----\n\n";
    }
    struct setupper
    {
        setupper()
        {
            if(iostream_untie)
            {
                std::ios::sync_with_stdio(false);
                std::cin.tie(nullptr);
            }
            std::cout << std::fixed << std::setprecision(__precision__);
#ifdef stderr_path
            if(freopen(stderr_path, "a", stderr))
            {
                std::cerr << std::fixed << std::setprecision(__precision__);
            }
#endif
#ifdef stdout_path
            if(not freopen(stdout_path, "w", stdout))
            {
                freopen("CON", "w", stdout);
                std::cerr << "Failed to open the stdout file\n\n";
            }
            std::cout << "";
#endif
#ifdef stdin_path
            if(not freopen(stdin_path, "r", stdin))
            {
                freopen("CON", "r", stdin);
                std::cerr << "Failed to open the stdin file\n\n";
            }
#endif
#ifdef LOCAL
            std::cerr << "----- stderr at LOCAL -----\n\n";
            atexit(print_elapsed_time);
            start_time = std::chrono::system_clock::now();
#else
            fclose(stderr);
#endif
        }
    } __setupper;
} // namespace execution

class myclock_t
{
    std::chrono::system_clock::time_point built_pt, last_pt;
    int built_ln, last_ln;
    std::string built_func, last_func;
    bool is_built;

  public:
    explicit myclock_t() : is_built(false)
    {
    }
    void build(int crt_ln, const std::string &crt_func)
    {
        is_built = true;
        last_pt = built_pt = std::chrono::system_clock::now();
        last_ln = built_ln = crt_ln, last_func = built_func = crt_func;
    }
    void set(int crt_ln, const std::string &crt_func)
    {
        if(is_built)
        {
            last_pt = std::chrono::system_clock::now();
            last_ln = crt_ln, last_func = crt_func;
        }
        else
        {
            debug_stream << "[ " << crt_ln << " : " << crt_func << " ] "
                         << "myclock_t::set failed (yet to be built!)\n";
        }
    }
    void get(int crt_ln, const std::string &crt_func)
    {
        if(is_built)
        {
            std::chrono::system_clock::time_point crt_pt(
                std::chrono::system_clock::now());
            int64_t diff =
                std::chrono::duration_cast<std::chrono::milliseconds>(crt_pt -
                                                                      last_pt)
                    .count();
            debug_stream << diff << " ms elapsed from"
                         << " [ " << last_ln << " : " << last_func << " ]";
            if(last_ln == built_ln) debug_stream << " (when built)";
            debug_stream << " to"
                         << " [ " << crt_ln << " : " << crt_func << " ]"
                         << "\n";
            last_pt = built_pt, last_ln = built_ln, last_func = built_func;
        }
        else
        {
            debug_stream << "[ " << crt_ln << " : " << crt_func << " ] "
                         << "myclock_t::get failed (yet to be built!)\n";
        }
    }
};
#ifdef LOCAL
myclock_t __myclock;
#define build_clock() __myclock.build(__LINE__, __func__)
#define set_clock() __myclock.set(__LINE__, __func__)
#define get_clock() __myclock.get(__LINE__, __func__)
#else
#define build_clock() ((void)0)
#define set_clock() ((void)0)
#define get_clock() ((void)0)
#endif

namespace std
{
    template <class RAitr>
    void rsort(RAitr __first, RAitr __last)
    {
        sort(__first, __last, greater<>());
    }
    template <class T>
    size_t hash_combine(size_t seed, T const &key)
    {
        return seed ^ (hash<T>()(key) + 0x9e3779b9 + (seed << 6) + (seed >> 2));
    }
    template <class T, class U>
    struct hash<pair<T, U>>
    {
        size_t operator()(pair<T, U> const &pr) const
        {
            return hash_combine(hash_combine(0, pr.first), pr.second);
        }
    };
    template <class tuple_t, size_t index = tuple_size<tuple_t>::value - 1>
    struct tuple_hash_calc
    {
        static size_t apply(size_t seed, tuple_t const &t)
        {
            return hash_combine(
                tuple_hash_calc<tuple_t, index - 1>::apply(seed, t),
                get<index>(t));
        }
    };
    template <class tuple_t>
    struct tuple_hash_calc<tuple_t, 0>
    {
        static size_t apply(size_t seed, tuple_t const &t)
        {
            return hash_combine(seed, get<0>(t));
        }
    };
    template <class... T>
    struct hash<tuple<T...>>
    {
        size_t operator()(tuple<T...> const &t) const
        {
            return tuple_hash_calc<tuple<T...>>::apply(0, t);
        }
    };
    template <class T, class U>
    istream &operator>>(std::istream &s, pair<T, U> &p)
    {
        return s >> p.first >> p.second;
    }
    template <class T, class U>
    ostream &operator<<(std::ostream &s, const pair<T, U> p)
    {
        return s << p.first << " " << p.second;
    }
    template <class T>
    istream &operator>>(istream &s, vector<T> &v)
    {
        for(T &e : v)
        {
            s >> e;
        }
        return s;
    }
    template <class T>
    ostream &operator<<(ostream &s, const vector<T> &v)
    {
        bool is_front = true;
        for(const T &e : v)
        {
            if(not is_front)
            {
                s << ' ';
            }
            else
            {
                is_front = false;
            }
            s << e;
        }
        return s;
    }
    template <class tuple_t, size_t index>
    struct tupleos
    {
        static ostream &apply(ostream &s, const tuple_t &t)
        {
            tupleos<tuple_t, index - 1>::apply(s, t);
            return s << " " << get<index>(t);
        }
    };
    template <class tuple_t>
    struct tupleos<tuple_t, 0>
    {
        static ostream &apply(ostream &s, const tuple_t &t)
        {
            return s << get<0>(t);
        }
    };
    template <class... T>
    ostream &operator<<(ostream &s, const tuple<T...> &t)
    {
        return tupleos<tuple<T...>, tuple_size<tuple<T...>>::value - 1>::apply(
            s, t);
    }
    template <>
    ostream &operator<<(ostream &s, const tuple<> &t)
    {
        return s;
    }
    string revstr(string str)
    {
        reverse(str.begin(), str.end());
        return str;
    }
} // namespace std

#ifdef LOCAL
#define dump(...)                                                              \
    debug_stream << "[ " << __LINE__ << " : " << __FUNCTION__ << " ]\n",       \
        dump_func(#__VA_ARGS__, __VA_ARGS__)
template <class T>
void dump_func(const char *ptr, const T &x)
{
    debug_stream << '\t';
    for(char c = *ptr; c != '\0'; c = *++ptr)
    {
        if(c != ' ') debug_stream << c;
    }
    debug_stream << " : " << x << '\n';
}
template <class T, class... rest_t>
void dump_func(const char *ptr, const T &x, rest_t... rest)
{
    debug_stream << '\t';
    for(char c = *ptr; c != ','; c = *++ptr)
    {
        if(c != ' ') debug_stream << c;
    }
    debug_stream << " : " << x << ",\n";
    dump_func(++ptr, rest...);
}
#else
#define dump(...) ((void)0)
#endif
template <class P>
void read_range(P __first, P __second)
{
    for(P i = __first; i != __second; ++i)
        std::cin >> *i;
}
template <class P>
void write_range(P __first, P __second)
{
    for(P i = __first; i != __second;
        std::cout << (++i == __second ? '\n' : ' '))
    {
        std::cout << *i;
    }
}

// substitute y for x.
template <class T>
void subst(T &x, const T &y)
{
    x = y;
}
// substitue y for x iff x > y.
template <class T>
bool chmin(T &x, const T &y)
{
    return x > y ? x = y, true : false;
}
// substitue y for x iff x < y.
template <class T>
bool chmax(T &x, const T &y)
{
    return x < y ? x = y, true : false;
}
template <class T>
constexpr T minf(const T &x, const T &y)
{
    return std::min(x, y);
}
template <class T>
constexpr T maxf(const T &x, const T &y)
{
    return std::max(x, y);
}
// binary search.
template <class int_t, class F>
int_t bin(int_t ok, int_t ng, const F &f)
{
    while(std::abs(ok - ng) > 1)
    {
        int_t mid = (ok + ng) / 2;
        (f(mid) ? ok : ng) = mid;
    }
    return ok;
}
// be careful that val is type-sensitive.
template <class T, class A, size_t N>
void init(A (&array)[N], const T &val)
{
    std::fill((T *)array, (T *)(array + N), val);
}
void reset()
{
}
template <class A, class... rest_t>
void reset(A &array, rest_t... rest)
{
    memset(array, 0, sizeof(array));
    reset(rest...);
}
// a integer uniformly and randomly chosen from the interval [l, r).
template <typename int_t>
int_t rand_int(int_t l, int_t r)
{
    static std::random_device seed_gen;
    static std::mt19937 engine(seed_gen());
    std::uniform_int_distribution<int_t> unid(l, r - 1);
    return unid(engine);
}
// a real number uniformly and randomly chosen from the interval [l, r).
template <typename real_t>
real_t rand_real(real_t l, real_t r)
{
    static std::random_device seed_gen;
    static std::mt19937 engine(seed_gen());
    std::uniform_real_distribution<real_t> unid(l, r);
    return unid(engine);
}

/* The main code follows. */

namespace math
{
    template <int mod>
    struct modint
    {
        int rep;

        constexpr modint() : rep(0)
        {}
        constexpr modint(int_fast64_t y)
            : rep(y >= 0 ? y % mod : (mod - (-y) % mod) % mod)
        {}

        constexpr modint &operator+=(const modint &p)
        {
            return (rep += p.rep) < mod ? 0 : rep -= mod, *this;
        }

        constexpr modint &operator++()
        {
            return ++rep, *this;
        }

        constexpr modint operator++(int)
        {
            modint t = *this;
            return ++rep, t;
        }

        constexpr modint &operator-=(const modint &p)
        {
            return (rep += mod - p.rep) < mod ? 0 : rep -= mod, *this;
        }

        constexpr modint &operator--()
        {
            return --rep, *this;
        }

        constexpr modint operator--(int)
        {
            modint t = *this;
            return --rep, t;
        }

        constexpr modint &operator*=(const modint &p)
        {
            return rep = (int_fast64_t)rep * p.rep % mod, *this;
        }

        constexpr modint &operator/=(const modint &p)
        {
            return *this *= inverse(p);
        }

        // constexpr modint &operator%=(int m) { return rep %= m, *this; }

        constexpr modint operator-() const
        {
            return modint(-rep);
        }

        constexpr modint operator+(const modint &p) const
        {
            return modint(*this) += p;
        }

        constexpr modint operator-(const modint &p) const
        {
            return modint(*this) -= p;
        }

        constexpr modint operator*(const modint &p) const
        {
            return modint(*this) *= p;
        }

        constexpr modint operator/(const modint &p) const
        {
            return modint(*this) /= p;
        }

        // constexpr modint operator%(int m) const { return modint(*this) %= m;
        // }

        constexpr bool operator==(const modint &p) const
        {
            return rep == p.rep;
        }

        constexpr bool operator!=(const modint &p) const
        {
            return rep != p.rep;
        }

        constexpr bool operator!() const
        {
            return !rep;
        }

        // constexpr bool operator>(const modint &p) const { return rep > p.rep; }

        // constexpr bool operator<(const modint &p) const { return rep <  p.rep; }

        // constexpr bool operator>=(const modint &p) const { return rep >= p.rep; }

        // constexpr bool operator<=(const modint &p) const { return rep <= p.rep; }

        constexpr friend modint<mod> inverse(const modint<mod> &p)
        {
            int a = p.rep, b = mod, u = 1, v = 0;
            while(b > 0)
            {
                int t = a / b;
                a -= t * b;
                a ^= b ^= a ^= b;
                u -= t * v;
                u ^= v ^= u ^= v;
            }
            return modint(u);
        }

        constexpr friend modint pow(modint p, int_fast64_t e)
        {
            if(e < 0) e = (e % (mod - 1) + mod - 1) % (mod - 1);
            modint ret = 1;
            while(e)
            {
                if(e & 1) ret *= p;
                p *= p;
                e >>= 1;
            }
            return ret;
        }

        friend std::ostream &operator<<(std::ostream &s, const modint &p)
        {
            return s << p.rep;
        }

        friend std::istream &operator>>(std::istream &s, modint &p)
        {
            int_fast64_t rep;
            p = modint((s >> rep, rep));
            return s;
        }
    };
} // namespace math

using namespace std;
using namespace math;

signed main()
{
    void __solve();
    void __precalc();

    unsigned int t = 1;
    // cin >> t;
    __precalc();

#ifdef LOCAL
    t = 1;
#endif

    while(t--)
    {
        __solve();
    }
}

using mint=modint<1000000007>;

void __precalc()
{
}

void __solve()
{
    int n, k;
    cin >> n >> k;
    mint dp[2][3125][55];
    reset(dp);
    auto &crt=dp[0];
    auto &nxt=dp[1];
    dp[0][0][0]=1;
    for(int i = 0; i < n; ++i)
    {
        reset(nxt);
        for(int s = 0; s <= n*n; ++s)
        {
            for(int j = 0; j < n; ++j)
            {
                if(not crt[s][j]) continue;
                nxt[s+j*2][j]+=crt[s][j]*(j*2+1);
                nxt[s+(j+1)*2][j+1]+=crt[s][j];
                if(j)
                {
                    nxt[s+(j-1)*2][j-1]+=crt[s][j]*j*j;
                }
            }
        }
        swap(crt,nxt);
    }
    std::cout << crt[k][0] << "\n";
}