A395447 Numerators of the coefficient of Pi^(4n+1) in Plouffe's formula for zeta(4n+1).

1, 125, 89, 397549, 68529640373, 2598638688071, 162580897794660958, 676470671886391879633, 38065792685808703577, 53866969189211783266383835533253, 1074151540472820600753617135934307286, 55141284330294633162607354950945193883

Offset

1,2

Links

Jwalin Bhatt, Table of n, a(n) for n = 1..136

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

a(n) = A394976(2n).

a(n)/A395448(n) = (2^(4n+1) / ((4n+2)! * (1 + (-4)^n - 2^(4n+1)))) * (Sum_{j=0..n} (-4)^(n+j) * binomial(4n+2, 4j) * B_{4j} * B_{4n-4j+2} + (1/2) * Sum_{j=0..2n+1} (-4)^j * binomial(4n+2, 2j) * B_{2j} * B_{4n-2j+2}), where B_k are the Bernoulli numbers (A027641/A027642).

Limit_{n->oo} (Pi^(4n+1)*a(n)/A395448(n)-coth(Pi)) / (zeta(4n+1)-1) = 2*coth(Pi)-coth(2*Pi).

Mathematica

Numerator[Table[2^(4n+1) (Sum[(-4)^(n+j) Binomial[4n+2, 4j] BernoulliB[4n-4j+2] BernoulliB[4j], {j, 0, n}] + 1/2 Sum[(-4)^j Binomial[4n+2, 2j] BernoulliB[4n-2j+2] BernoulliB[2j], {j, 0, 2n+1}]) / ((4n+2)! (1 + (-4)^n - 2^(4n+1))), {n, 1, 10}]]

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)
    )
    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, A394976, A395448.

Sequence in context: A030668 A030678 A028939 * A195420 A080538 A104233

Adjacent sequences: A395444 A395445 A395446 * A395448 A395449 A395450

Keyword

nonn,frac

Author

Jwalin Bhatt, Apr 23 2026

Status

approved