OFFSET
1,1
LINKS
Jwalin Bhatt, Table of n, a(n) for n = 1..112
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) = A394977(2n).
A395447(n)/a(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)*A395447(n)/a(n)-coth(Pi)) / (zeta(4n+1)-1) = 2*coth(Pi)-coth(2*Pi).
MATHEMATICA
Denominator[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.denominator
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Jwalin Bhatt, Apr 23 2026
STATUS
approved