135 lines
3.2 KiB
C
135 lines
3.2 KiB
C
/*=============================================================================
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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) 2010 Fredrik Johansson
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******************************************************************************/
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#include "arith.h"
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static __inline__ void
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_fmpz_addmul_alt(fmpz_t s, fmpz_t t, fmpz_t u, int parity)
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{
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if (parity % 2)
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fmpz_submul(s, t, u);
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else
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fmpz_addmul(s, t, u);
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}
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static void
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_fmpz_stirling2_powsum(fmpz_t s, slong n, slong k)
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{
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fmpz_t t, u;
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fmpz * bc;
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slong j, m, max_bc;
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fmpz_init(t);
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fmpz_init(u);
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max_bc = (k+1) / 2;
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bc = _fmpz_vec_init(max_bc + 1);
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fmpz_one(bc);
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for (j = 1; j <= max_bc; j++)
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{
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fmpz_set(bc+j, bc+j-1);
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fmpz_mul_ui(bc+j, bc+j, k+1-j);
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fmpz_divexact_ui(bc+j, bc+j, j);
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}
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fmpz_zero(s);
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for (j = 1; j <= k; j += 2)
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{
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fmpz_set_ui(u, j);
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fmpz_pow_ui(u, u, n);
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m = j;
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/* Process each m = 2^p * j */
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while (1)
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{
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if (m > max_bc)
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_fmpz_addmul_alt(s, bc+k-m, u, k + m);
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else
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_fmpz_addmul_alt(s, bc+m, u, k + m);
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m *= 2;
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if (m > k)
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break;
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fmpz_mul_2exp(u, u, n);
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}
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}
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_fmpz_vec_clear(bc, max_bc + 1);
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fmpz_fac_ui(t, k);
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fmpz_divexact(s, s, t);
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fmpz_clear(t);
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fmpz_clear(u);
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}
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void
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arith_stirling_number_2(fmpz_t s, slong n, slong k)
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{
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if (n < 0 || k < 0 || k > n)
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{
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fmpz_zero(s);
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return;
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}
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/* Topmost diagonals */
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if (k >= n - 1)
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{
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if (k == n)
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fmpz_one(s);
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else /* k == n - 1 */
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{
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/* S(n,n-1) = binomial(n,2) */
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fmpz_set_ui(s, n);
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fmpz_mul_ui(s, s, n-1);
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fmpz_divexact_ui(s, s, UWORD(2));
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}
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return;
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}
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/* Leftmost columns */
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if (k <= 2)
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{
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if (k < 2)
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fmpz_set_ui(s, k);
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else
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{
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/* S(n,2) = 2^(n-1)-1 */
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fmpz_one(s);
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fmpz_mul_2exp(s, s, n-1);
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fmpz_sub_ui(s, s, UWORD(1));
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}
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return;
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}
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_fmpz_stirling2_powsum(s, n, k);
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}
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void
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arith_stirling_number_2_vec(fmpz * row, slong n, slong klen)
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{
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slong m;
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for (m = 0; m <= n; m++)
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arith_stirling_number_2_vec_next(row, row, m, klen);
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}
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