A320419 E.g.f.: Sum_{n>=0} 2^n * sinh(n*x)^n.

1, 2, 32, 1298, 98816, 12116642, 2181373952, 541793612978, 177515752718336, 74174630255081282, 38495436789222735872, 24292625097918019749458, 18317925825330618728185856, 16266073932645598088605425122, 16800468023465020621665905672192, 19969924961381649826994229325322738

Offset: 0

Author

Paul D. Hanna, Oct 14 2018

Keywords

• nonn

Comments

Given e.g.f. A(x),

(1) A(log(1+x)) is the g.f. of A319466,

(1) A(-log(1-x)) is the g.f. of A319947.

Examples

			E.g.f.: A(x) = 1 + 2*x + 32*x^2/2! + 1298*x^3/3! + 98816*x^4/4! + 12116642*x^5/5! + 2181373952*x^6/6! + 541793612978*x^7/7! + ...
such that
A(x) = 1 + 2*sinh(x) + 4*sinh(2*x)^2 + 8*sinh(3*x)^3 + 16*sinh(4*x)^4 +...
or, equivalently,
A(x) = 1 + exp(x)*(1 - exp(-2*x)) + exp(4*x)*(1 - exp(-4*x))^2 + exp(9*x)*(1 - exp(-6*x))^3 + exp(16*x)*(1 - exp(-8*x))^4 + exp(25*x)*(1 - exp(-10*x))^5 + ...
RELATED SERIES.
A(log(1+x)) = 1 + 2*x + 15*x^2 + 201*x^3 + 3807*x^4 + 93103*x^5 + 2788528*x^6 + 98816388*x^7 + 4043274742*x^8 + ... + A319466(n)*x^n + ...
A(-log(1-x)) = 1 + 2*x + 17*x^2 + 233*x^3 + 4457*x^4 + 109599*x^5 + 3294200*x^6 + 117023348*x^7 + 4796944724*x^8 + ... + A319947(n)*x^n + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A224899, A319466, A319947.

Programs

PARI

{a(n) = n! * polcoeff(sum(k=0, n, 2^k * sinh(k*x + x*O(x^n))^k ), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: Sum_{n>=0} exp(n^2*x) * (1 - exp(-2*n*x))^n.

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - exp(x + y) + exp(x - y)). - Ilya Gutkovskiy, Apr 24 2025

a(n) ~ n!^2 * c * d^n / sqrt(n), where d = 5.4666049332127684665699843922982444983683628264... and c = 0.390468512121689057564560997910519445284386310369... - Vaclav Kotesovec, Apr 24 2025