A330905 Numerator of 1/Pi^(4*n+3) * Sum_{k>0} (-1)^(k+1) / (k^(4*n+3) * sinh(Pi * k)).

1, 13, 4009, 13739, 26190337, 790092547807, 15121363327643, 2442193001593535677, 41701392468830919939353, 17185858614142258665062467, 17099921279612182344285033157, 28295511786898541163838004665601843, 727977487532189566289706245511979571, 22681093492188834346091000609641534617709

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..100

MathStackexchange, A Ramanujan sum involving sinh

Wikipedia, Bernoulli number

Formula

Let B_n be the Bernoulli number.

a(n)/A330906(n) = Sum_{k=0..2*n+2} (-1)^k*(1-2^(2*k-1))*(1-2^(4*n+3-2*k))*B_{2*k}*B_{4*n+4-2*k}/((2*k)!*(4*n+4-2*k)!)).

Example

1/360, 13/453600, 4009/13621608000, 13739/4547140416000, 26190337/844351508246400000, 790092547807/2481187700290640140800000, ... = a(n)/A330906(n).

Mathematica

a[n_] := Numerator[Sum[(-1)^k * (1 - 2^(2*k - 1)) * (1 - 2^(4*n + 3 - 2*k)) * BernoulliB[2*k] * BernoulliB[4*n + 4 - 2*k]/((2*k)!*(4*n + 4 - 2*k)!), {k, 0, 2*n + 2}]]; Array[a, 14, 0] (* Amiram Eldar, May 01 2020 *)

Prog

(PARI) {a(n) = numerator(sum(k=0, 2*n+2, (-1)^k*(1-2^(2*k-1))*(1-2^(4*n+3-2*k))*bernfrac(2*k)*bernfrac(4*n+4-2*k)/((2*k)!*(4*n+4-2*k)!)))}

Crossrefs

Cf. A004767, A057866/A057867, A330906 (denominator).

Sequence in context: A337147 A070905 A180768 * A068532 A182348 A094316

Adjacent sequences: A330902 A330903 A330904 * A330906 A330907 A330908

Keyword

nonn,frac

Author

Seiichi Manyama, May 01 2020

Status

approved