1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
#include <vector>
#include <iostream>
#include <cassert>
using namespace std;

#define NDEBUG
#define LAZY 0

using ll = long long;

ll fpw(ll a, ll b, ll p){
    ll r = 1;
    while(b) {
        if(b&1) r = r*a%p;
        a = a*a%p; b /= 2;
    }
    return r;
}

ll M;

vector<ll> fac;
vector<ll> fac_inv;
vector<ll> Vcompute_range;
vector<ll> Vcompute;
vector<vector<ll> > Vnewton;

inline ll AM(ll a) {
    //if (a > M) a -= M; return a;
    return a % M;
}

const int MLIMIT = 12;

int I(ll n, ll m, bool sl, bool sr) {
    assert(n >= 0 && m >= 0);
    assert(m < MLIMIT && n <= 1000);
    //return (MLIMIT*n + m) * 4 + (sl?1:0) + (sr?2:0);
    return (MLIMIT*n + m) + ((sl?1:0) + (sr?2:0))*1001*MLIMIT;
    //return (n + m*1001) + ((sl?1:0) + (sr?2:0))*1001*MLIMIT;
}

ll newton(ll n, ll k) {
    return Vnewton[n][k];//(((fac[n] * fac_inv[k]) % M) * fac_inv[n-k]) % M;
}

ll do_compute_range(ll n, ll m, ll sl, ll sr);
ll do_compute(ll n, ll m, ll sl, ll sr);

ll compute_range(ll n, ll m, ll sl, ll sr) {
    if (m <= 0) return 0;
    #if LAZY
    if (Vcompute_range[I(n,m,sl,sr)] == -1) Vcompute_range[I(n,m,sl,sr)] = do_compute_range(n,m,sl,sr);
    #endif
    return Vcompute_range[I(n,m,sl,sr)];
}

ll compute(ll n, ll m, ll sl, ll sr) {
    if (m < 0) return 0;
    #if LAZY
    if (Vcompute[I(n,m,sl,sr)] == -1) Vcompute[I(n,m,sl,sr)] = do_compute(n,m,sl,sr);
    #endif
    return Vcompute[I(n,m,sl,sr)];
}

ll do_compute_range(ll n, ll m, ll sl, ll sr) {
    if(m <= 0) return 0;
    return AM(compute(n, m-1, sl, sr) + compute_range(n, m-1, sl, sr));
}

ll do_compute(ll n, ll m, ll sl, ll sr) {
    if(n == 0) {
        if(m == 0) {
            return 1;
        } else {
            return 0;
        }
    }

    if(m < 0) return 0;

    ll res = 0;
    bool fastmode = sl == sr;
    for(ll k1=0; k1 < (fastmode ? ((n+1)/2) : n); k1++) {
        ll k2 = n - 1 - k1;
        ll x = newton(n-1, k1);

        ll lres = 0;

        // both finish at the same time
        lres += (compute(k1, m-1, sl, true) * compute(k2, m-1, true, sr)) % M;
        //lres %= M;

        if(sl) {
            // left dies first, we are consumed by wall
            lres += (compute_range(k1, m, sl, true) * compute(k2, m, true, sr)) % M;
        } else {
            // left dies first, no wall
            lres += (compute_range(k1, m-1, sl, true) * compute(k2, m-1, true, sr)) % M;
        }
        //lres %= M;
        if(sr) {
            // right dies first, we are consumed by wall
            lres += (compute(k1, m, sl, true) * compute_range(k2, m, true, sr)) % M;
        } else {
            // right dies first, no wall
            lres += (compute(k1, m-1, sl, true) * compute_range(k2, m-1, true, sr)) % M;
        }

        if (fastmode && 2*k1+1 != n) {
            lres *= 2;
        }

        lres %= M;

        //print(n,m, '..', k1, k2, '->', lres)
        res += (lres * x);
        res %= M;
    }

    return res;
}

bool debug = false;

int main() {
    int n, k;
    cin >> n >> k >> M;
    if (M == -1) { debug = true; M = 100000007; }

    Vcompute.resize(1001 * MLIMIT * 4, -1);
    Vcompute_range.resize(1001 * MLIMIT * 4, -1);

    fac.push_back(1);
    fac_inv.push_back(1);
    for (int i=1; i <= 1005; i ++) {
        fac.push_back((fac.back() * i) % M);
        fac_inv.push_back(fpw(fac.back(), M-2, M));
    }

    Vnewton.resize(1001, vector<ll>(1001));
    for (int n=0; n < 1001; n++)
        for (int k=0; k < 1001; k++)
            Vnewton[n][k] = (((fac[n] * fac_inv[k]) % M) * fac_inv[n-k]) % M;

    #if !LAZY
    for (ll n=0; n <= 1000; n++ ){
        for (ll m=0; m < MLIMIT; m++ ){
            for (int sl=0; sl <= 1; sl ++) {
                for (int sr=0; sr <= 1; sr ++) {
                    Vcompute[I(n,m, sl, sr)] = do_compute(n,m, sl, sr);
                    Vcompute_range[I(n,m, sl, sr)] = do_compute_range(n,m, sl, sr);
                }
            }
        }
    }
    #endif

    if (debug) {
        for (int n=0; n <= 1000; n ++){
            cout << "[";
            for (int m=0; m < MLIMIT; m ++){
                if(m) cout << ", ";
                cout << compute(n, m, false, false);
            }
            cout << "]" << endl;
        }
        cout << compute(1000, 10 + 1, false, false) << endl;

        ll chksum = 0;

        for (int n=0; n <= 1000; n ++){
            for (int m=1; m < MLIMIT; m ++){
                chksum += (compute(n, m, false, false) * n * (m-1)) % M;
                chksum %= M;
            }
        }
        cout << chksum << endl;
        assert(chksum == 59180009);
    } else {
        k ++;
        if (k >= MLIMIT) {
            cout << 0 << endl;
        } else {
            cout << compute(n, k, false, false) << endl;
        }
    }
}