A395447 - OEIS

%I #20 Apr 28 2026 00:08:55

%S 1,125,89,397549,68529640373,2598638688071,162580897794660958,

%T 676470671886391879633,38065792685808703577,

%U 53866969189211783266383835533253,1074151540472820600753617135934307286,55141284330294633162607354950945193883

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

%H Jwalin Bhatt, <a href="/A395447/b395447.txt">Table of n, a(n) for n = 1..136</a>

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/inspired2.pdf">Identities inspired by Ramanujan Notebooks (part 2)</a>, April 2006.

%H Linas Vepštas, <a href="https://doi.org/10.1007/s11139-011-9335-9">On Plouffe's Ramanujan identities</a>, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; <a href="https://arxiv.org/abs/math/0609775">arXiv preprint</a>, arXiv:math/0609775 [math.NT], 2006-2010.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RiemannZetaFunction.html">Riemann Zeta Function</a>.

%F a(n) = A394976(2n).

%F 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).

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

%t 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}]]

%o (Python)

%o from sympy import binomial, bernoulli, factorial

%o def a(n):

%o     s1 = sum(

%o         (-4)**(n+j) * binomial(4*n+2, 4*j) * bernoulli(4*n-4*j+2) * bernoulli(4*j)

%o         for j in range(n+1)

%o     )

%o     s2 = sum(

%o         (-4)**j * binomial(4*n+2, 2*j) * bernoulli(4*n-2*j+2) * bernoulli(2*j)

%o         for j in range(2*n+2)

%o     )

%o     frac = 2**(4*n+1) * (s1+s2/2) / factorial(4*n+2) / (1 + (-4)**n - 2**(4*n+1))

%o     return frac.numerator

%Y Cf. A057866, A394976, A395448.

%K nonn,frac

%O 1,2

%A _Jwalin Bhatt_, Apr 23 2026