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+/* bernoulli -- internal function to compute Bernoulli numbers.
+
+Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
+Contributed by the AriC and Caramel projects, INRIA.
+
+This file is part of the GNU MPFR Library.
+
+The GNU MPFR Library is free software; you can redistribute it and/or modify
+it under the terms of the GNU Lesser General Public License as published by
+the Free Software Foundation; either version 3 of the License, or (at your
+option) any later version.
+
+The GNU MPFR Library 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 Lesser General Public
+License for more details.
+
+You should have received a copy of the GNU Lesser General Public License
+along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
+http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
+51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
+
+#include "mpfr-impl.h"
+
+/* assuming b[0]...b[2(n-1)] are computed, computes and stores B[2n]*(2n+1)!
+
+ t/(exp(t)-1) = sum(B[j]*t^j/j!, j=0..infinity)
+ thus t = (exp(t)-1) * sum(B[j]*t^j/j!, n=0..infinity).
+ Taking the coefficient of degree n+1 > 1, we get:
+ 0 = sum(1/(n+1-k)!*B[k]/k!, k=0..n)
+ which gives:
+ B[n] = -sum(binomial(n+1,k)*B[k], k=0..n-1)/(n+1).
+
+ Let C[n] = B[n]*(n+1)!.
+ Then C[n] = -sum(binomial(n+1,k)*C[k]*n!/(k+1)!, k=0..n-1),
+ which proves that the C[n] are integers.
+*/
+mpz_t*
+mpfr_bernoulli_internal (mpz_t *b, unsigned long n)
+{
+ if (n == 0)
+ {
+ b = (mpz_t *) (*__gmp_allocate_func) (sizeof (mpz_t));
+ mpz_init_set_ui (b[0], 1);
+ }
+ else
+ {
+ mpz_t t;
+ unsigned long k;
+
+ b = (mpz_t *) (*__gmp_reallocate_func)
+ (b, n * sizeof (mpz_t), (n + 1) * sizeof (mpz_t));
+ mpz_init (b[n]);
+ /* b[n] = -sum(binomial(2n+1,2k)*C[k]*(2n)!/(2k+1)!, k=0..n-1) */
+ mpz_init_set_ui (t, 2 * n + 1);
+ mpz_mul_ui (t, t, 2 * n - 1);
+ mpz_mul_ui (t, t, 2 * n);
+ mpz_mul_ui (t, t, n);
+ mpz_fdiv_q_ui (t, t, 3); /* exact: t=binomial(2*n+1,2*k)*(2*n)!/(2*k+1)!
+ for k=n-1 */
+ mpz_mul (b[n], t, b[n-1]);
+ for (k = n - 1; k-- > 0;)
+ {
+ mpz_mul_ui (t, t, 2 * k + 1);
+ mpz_mul_ui (t, t, 2 * k + 2);
+ mpz_mul_ui (t, t, 2 * k + 2);
+ mpz_mul_ui (t, t, 2 * k + 3);
+ mpz_fdiv_q_ui (t, t, 2 * (n - k) + 1);
+ mpz_fdiv_q_ui (t, t, 2 * (n - k));
+ mpz_addmul (b[n], t, b[k]);
+ }
+ /* take into account C[1] */
+ mpz_mul_ui (t, t, 2 * n + 1);
+ mpz_fdiv_q_2exp (t, t, 1);
+ mpz_sub (b[n], b[n], t);
+ mpz_neg (b[n], b[n]);
+ mpz_clear (t);
+ }
+ return b;
+}