cp's OEIS Frontend

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

%I A320419 #16 Apr 24 2025 06:53:26
%S A320419 1,2,32,1298,98816,12116642,2181373952,541793612978,177515752718336,
%T A320419 74174630255081282,38495436789222735872,24292625097918019749458,
%U A320419 18317925825330618728185856,16266073932645598088605425122,16800468023465020621665905672192,19969924961381649826994229325322738
%N A320419 E.g.f.: Sum_{n>=0} 2^n * sinh(n*x)^n.
%C A320419 Given e.g.f. A(x),
%C A320419 (1) A(log(1+x)) is the g.f. of A319466,
%C A320419 (1) A(-log(1-x)) is the g.f. of A319947.
%H A320419 Paul D. Hanna, <a href="/A320419/b320419.txt">Table of n, a(n) for n = 0..200</a>
%F A320419 E.g.f.: Sum_{n>=0} exp(n^2*x) * (1 - exp(-2*n*x))^n.
%F A320419 a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - exp(x + y) + exp(x - y)). - _Ilya Gutkovskiy_, Apr 24 2025
%F A320419 a(n) ~ n!^2 * c * d^n / sqrt(n), where d = 5.4666049332127684665699843922982444983683628264... and c = 0.390468512121689057564560997910519445284386310369... - _Vaclav Kotesovec_, Apr 24 2025
%e A320419 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! + ...
%e A320419 such that
%e A320419 A(x) = 1 + 2*sinh(x) + 4*sinh(2*x)^2 + 8*sinh(3*x)^3 + 16*sinh(4*x)^4 +...
%e A320419 or, equivalently,
%e A320419 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 + ...
%e A320419 RELATED SERIES.
%e A320419 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 + ...
%e A320419 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 + ...
%o A320419 (PARI) {a(n) = n! * polcoeff(sum(k=0, n, 2^k * sinh(k*x + x*O(x^n))^k ), n)}
%o A320419 for(n=0, 30, print1(a(n), ", "))
%Y A320419 Cf. A224899, A319466, A319947.
%K A320419 nonn
%O A320419 0,2
%A A320419 _Paul D. Hanna_, Oct 14 2018