111 lines
3.6 KiB
C
111 lines
3.6 KiB
C
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/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2011 Fredrik Johansson
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******************************************************************************/
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#include "arith.h"
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const mp_limb_t bell_number_tab[] =
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{
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UWORD(1), UWORD(1), UWORD(2), UWORD(5), UWORD(15), UWORD(52), UWORD(203), UWORD(877), UWORD(4140), UWORD(21147), UWORD(115975),
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UWORD(678570), UWORD(4213597), UWORD(27644437), UWORD(190899322), UWORD(1382958545),
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#if FLINT64
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UWORD(10480142147), UWORD(82864869804), UWORD(682076806159), UWORD(5832742205057),
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UWORD(51724158235372), UWORD(474869816156751), UWORD(4506715738447323),
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UWORD(44152005855084346), UWORD(445958869294805289),
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UWORD(4638590332229999353),
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#endif
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};
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static const char bell_mod_2[3] = {1, 1, 0};
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static const char bell_mod_3[13] = {1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1};
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mp_limb_t
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arith_bell_number_nmod(ulong n, nmod_t mod)
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{
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mp_limb_t s, t, u;
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mp_ptr facs, pows;
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slong i, j;
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if (n < BELL_NUMBER_TAB_SIZE)
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return n_mod2_preinv(bell_number_tab[n], mod.n, mod.ninv);
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if (mod.n == 2) return bell_mod_2[n % 3];
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if (mod.n == 3) return bell_mod_3[n % 13];
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if (mod.n <= n)
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{
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mp_ptr bvec = flint_malloc(sizeof(mp_limb_t) * (n + 1));
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arith_bell_number_nmod_vec_recursive(bvec, n + 1, mod);
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s = bvec[n];
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flint_free(bvec);
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return s;
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}
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/* Compute inverse factorials */
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/* We actually compute (n! / i!) and divide out (n!)^2 at the end */
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facs = flint_malloc(sizeof(mp_limb_t) * (n + 1));
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facs[n] = 1;
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for (i = n - 1; i >= 0; i--)
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facs[i] = n_mulmod2_preinv(facs[i + 1], i + 1, mod.n, mod.ninv);
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/* Compute powers */
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pows = flint_calloc(n + 1, sizeof(mp_limb_t));
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pows[0] = n_powmod2_ui_preinv(0, n, mod.n, mod.ninv);
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pows[1] = n_powmod2_ui_preinv(1, n, mod.n, mod.ninv);
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for (i = 2; i <= n; i++)
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{
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if (pows[i] == 0)
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pows[i] = n_powmod2_ui_preinv(i, n, mod.n, mod.ninv);
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for (j = 2; j <= i && i * j <= n; j++)
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if (pows[i * j] == 0)
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pows[i * j] = n_mulmod2_preinv(pows[i],
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pows[j], mod.n, mod.ninv);
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}
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for (s = t = i = 0; i <= n; i++)
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{
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if (i % 2 == 0)
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t = n_addmod(t, facs[i], mod.n);
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else
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t = n_submod(t, facs[i], mod.n);
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u = pows[n - i];
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u = n_mulmod2_preinv(u, facs[n - i], mod.n, mod.ninv);
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u = n_mulmod2_preinv(u, t, mod.n, mod.ninv);
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s = n_addmod(s, u, mod.n);
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}
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/* Remove (n!)^2 */
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u = n_invmod(facs[0], mod.n);
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u = n_mulmod2_preinv(u, u, mod.n, mod.ninv);
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s = n_mulmod2_preinv(s, u, mod.n, mod.ninv);
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flint_free(facs);
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flint_free(pows);
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return s;
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}
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