OFFSET
0,3
COMMENTS
Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, ...], with an apparent period of 6. Cf. A245322. - Peter Bala, May 29 2022
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..225
FORMULA
E.g.f.: Sum_{n>=0} exp(-n^2*x) * (exp(2*n*x) - 1)^n / 2^n.
a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / (sqrt(3-2*log(2)) * 3^(n+1/2) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Oct 28 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 8*x^2/2! + 163*x^3/3! + 6272*x^4/4! +...
where
A(x) = 1 + sinh(x) + sinh(2*x)^2 + sinh(3*x)^3 + sinh(4*x)^4 +...
MATHEMATICA
Flatten[{1, Table[Sum[Sum[Binomial[k, j] * (-1)^j * k^n*(k-2*j)^n / 2^k, {j, 0, k}], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 29 2014 *)
Join[{1}, Rest[With[{nn=20}, CoefficientList[Series[Sum[Sinh[n*x]^n, {n, nn}], {x, 0, nn}], x] Range[0, nn]!]]] (* Harvey P. Dale, May 18 2018 *)
PROG
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, sinh(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Paul D. Hanna, Jul 24 2013
STATUS
approved