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/*=============================================================================
This file is part of FLINT.
FLINT is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
FLINT is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FLINT; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2010 Fredrik Johansson
******************************************************************************/
#include "arith.h"
static __inline__ void
_fmpz_addmul_alt(fmpz_t s, fmpz_t t, fmpz_t u, int parity)
{
if (parity % 2)
fmpz_submul(s, t, u);
else
fmpz_addmul(s, t, u);
}
static void
_fmpz_stirling2_powsum(fmpz_t s, slong n, slong k)
{
fmpz_t t, u;
fmpz * bc;
slong j, m, max_bc;
fmpz_init(t);
fmpz_init(u);
max_bc = (k+1) / 2;
bc = _fmpz_vec_init(max_bc + 1);
fmpz_one(bc);
for (j = 1; j <= max_bc; j++)
{
fmpz_set(bc+j, bc+j-1);
fmpz_mul_ui(bc+j, bc+j, k+1-j);
fmpz_divexact_ui(bc+j, bc+j, j);
}
fmpz_zero(s);
for (j = 1; j <= k; j += 2)
{
fmpz_set_ui(u, j);
fmpz_pow_ui(u, u, n);
m = j;
/* Process each m = 2^p * j */
while (1)
{
if (m > max_bc)
_fmpz_addmul_alt(s, bc+k-m, u, k + m);
else
_fmpz_addmul_alt(s, bc+m, u, k + m);
m *= 2;
if (m > k)
break;
fmpz_mul_2exp(u, u, n);
}
}
_fmpz_vec_clear(bc, max_bc + 1);
fmpz_fac_ui(t, k);
fmpz_divexact(s, s, t);
fmpz_clear(t);
fmpz_clear(u);
}
void
arith_stirling_number_2(fmpz_t s, slong n, slong k)
{
if (n < 0 || k < 0 || k > n)
{
fmpz_zero(s);
return;
}
/* Topmost diagonals */
if (k >= n - 1)
{
if (k == n)
fmpz_one(s);
else /* k == n - 1 */
{
/* S(n,n-1) = binomial(n,2) */
fmpz_set_ui(s, n);
fmpz_mul_ui(s, s, n-1);
fmpz_divexact_ui(s, s, UWORD(2));
}
return;
}
/* Leftmost columns */
if (k <= 2)
{
if (k < 2)
fmpz_set_ui(s, k);
else
{
/* S(n,2) = 2^(n-1)-1 */
fmpz_one(s);
fmpz_mul_2exp(s, s, n-1);
fmpz_sub_ui(s, s, UWORD(1));
}
return;
}
_fmpz_stirling2_powsum(s, n, k);
}
void
arith_stirling_number_2_vec(fmpz * row, slong n, slong klen)
{
slong m;
for (m = 0; m <= n; m++)
arith_stirling_number_2_vec_next(row, row, m, klen);
}
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