A395748 Coefficient of S_p(n) in the linear equality by Plouffe for the odd values of the Riemann zeta function, where S_p(n) = Sum_{k>=1} 1/((k^n)*(exp(2*Pi*k)+1)).

84, 74844, 62370, 29116187100, 1793047592085750, 414193993771808250, 157739569618086594888750, 3995914450354646593461187500, 50649968252302318662731718750, 11793580064115475681690378028576697656250, 1431752413363682863232893583047239672166406250

Offset

1,1

Links

Jwalin Bhatt, Table of n, a(n) for n = 1..126

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.

Wikipedia, zeta(2n+1).

Formula

A395662(2n)*zeta(4n+1) - A395663(2n)*Pi^(4n+1) + A396050(2n)*S_m(4n+1) + a(n)*S_p(4n+1) = 0, where S_p(n) = Sum_{k>=1} 1/((k^n)*(exp(2*Pi*k)+1)) and S_m(n) = Sum_{k>=1} 1/((k^n)*(exp(2*Pi*k)-1)).

Example

1470*zeta(5) - 5*Pi^(5) + 3024*S_m(5) + 84*S_p(5) = 0.
18523890*zeta(9) - 625*Pi^(9) + 37122624*S_m(9) + 74844*S_p(9) = 0.
257432175*zeta(13) - 89*Pi^(13) + 514926720*S_m(13) + 62370*S_p(13) = 0.

Mathematica

a[n_] := Module[{s1, s2, f},
  s1 = Sum[
    (-4)^(n + j) * Binomial[4*n + 2, 4*j] * BernoulliB[4*n - 4*j + 2] * BernoulliB[4*j],
    {j, 0, n}
  ];
  s2 = Sum[
    (-4)^j * Binomial[4*n + 2, 2*j] * BernoulliB[4*n - 2*j + 2] * BernoulliB[2*j],
    {j, 0, 2*n + 1}
  ];
  f = 2^(4*n + 1) * (s1 + s2/2) / Factorial[4*n + 2];
  2 * Denominator[f]
]

Prog

(Python)
from sympy import binomial, bernoulli, factorial
def a(n):
    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)
    )
    f = 2**(4*n+1) * (s1+s2/2) / factorial(4*n+2)
    return 2*f.q

Crossrefs

Cf. A395662, A395663, A396050.

Sequence in context: A203093 A290274 A289557 * A202580 A292196 A076230

Adjacent sequences: A395745 A395746 A395747 * A395749 A395750 A395751

Keyword

nonn

Author

Jwalin Bhatt, May 05 2026

Status

approved