A394976 Numerators of the coefficient of Pi^(2n+1) in Plouffe's formula for zeta(2n+1).

7, 1, 19, 125, 1453, 89, 13687, 397549, 7708537, 68529640373, 4472029801, 2598638688071, 149780635937, 162580897794660958, 11231299844779783, 676470671886391879633, 3688053840923281541, 38065792685808703577, 2659842854283579394387

Offset

1,1

Links

Jwalin Bhatt, Table of n, a(n) for n = 1..273

Simon Plouffe, Identities inspired by Ramanujan Notebooks (part 2), April 2006.

Linas Vepštas, On Plouffe's Ramanujan identities, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; arXiv preprint, arXiv:math/0609775 [math.NT], 2006-2010.

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

Formula

Limit_{n->oo} (a(n)/A394977(n)) * Pi^(2n+1) = coth(Pi) (A175316).

a(n) = A057866((n+1)/2) for odd n.

a(n) = A395447(n/2) for even n. - Jwalin Bhatt, May 12 2026

Mathematica

a[n_] := Module[{n0 = n, s, s1, s2, frac},
  If[OddQ[n], (* 4n - 1 case *)
   n0 = Floor[n/2] + 1;
   s = Sum[
     (-1)^j * Binomial[4*n0, 2*j] * BernoulliB[4*n0 - 2*j] * BernoulliB[2*j],
     {j, 0, 2*n0}
   ];
   frac = -2^(4*n0 - 2) * s / Factorial[4*n0];
   ,
   (* 4n + 1 case *)
   n0 = Floor[n/2];
   s1 = Sum[
     (-4)^(n0 + j) * Binomial[4*n0 + 2, 4*j] * BernoulliB[4*n0 - 4*j + 2] * BernoulliB[4*j],
     {j, 0, n0}
   ];
   s2 = Sum[
     (-4)^j * Binomial[4*n0 + 2, 2*j] * BernoulliB[4*n0 - 2*j + 2] * BernoulliB[2*j],
     {j, 0, 2*n0 + 1}
   ];
   frac = 2^(4*n0 + 1) * (s1 + s2/2) / (Factorial[4*n0 + 2] * (1 + (-4)^n0 - 2^(4*n0 + 1)));
  ];
  Numerator[frac]
]

Prog

(Python)
from sympy import binomial, bernoulli, factorial
def a(n):
    if n%2 == 1: # 4n - 1
        n = n//2 + 1
        s = sum(
            (-1)**j * binomial(4*n, 2*j) * bernoulli(4*n-2*j) * bernoulli(2*j)
            for j in range(2*n + 1)
        )
        frac = -2**(4*n-2) * s / factorial(4*n)
    else: # 4n + 1
        n //= 2
        s1 = sum(
            (-4)**(n+j) * binomial(4*n+2, 4*j) * bernoulli(4*n-4*j+2) * bernoulli(4*j)
            for j in range(n+1)
        )
        s2 = sum(
            (-4)**j * binomial(4*n+2, 2*j) * bernoulli(4*n-2*j+2) * bernoulli(2*j)
            for j in range(2*n+2)
        )
        frac = 2**(4*n+1) * (s1+s2/2) / factorial(4*n+2) / (1 + (-4)**n - 2**(4*n+1))
    return frac.numerator

Crossrefs

Cf. A057866, A175316, A394977 (denominators), A395447.

Sequence in context: A215503 A371835 A195220 * A119546 A173204 A229820

Adjacent sequences: A394973 A394974 A394975 * A394977 A394978 A394979

Keyword

nonn,frac

Author

Jwalin Bhatt, Apr 08 2026

Status

approved