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
KEYWORD
nonn,frac
AUTHOR
Seiichi Manyama, May 01 2020
STATUS
approved