A381415 E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)^2) ).

1, 1, 5, 50, 765, 15852, 415441, 13182976, 491502521, 21061603152, 1020066862269, 55107133707232, 3285531022228725, 214295961023511616, 15179005200468020489, 1160334809344169734144, 95214513195493336071537, 8347897781857074205573376, 778804910740650550851809013

Offset: 0

Author

Seiichi Manyama, Feb 23 2025

Keywords

• nonn

Crossrefs

Cf. A136630, A162650.

Programs

Mathematica

Join[{1}, Table[Sum[(2*n + 1)^(k-1) / (2^k*k!) * Sum[(-1)^(k-j) * (2*j - k)^n * Binomial[k, j], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jul 04 2025 *)

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (2*n+1)^(k-1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * A136630(n,k).

a(n) ~ s * n^(n-1) / (2*sqrt(1 + r*s^2*sqrt(1 - 4*r^2*s^4)) * exp(n) * r^n), where r = 0.1774317812751606880070098054556619184142424898705... and s = 1.597465072615091018021826608474818660705268320323... are the roots of the system of equations exp(sinh(r*s^2)) = s, 2*r*s^2*cosh(r*s^2) = 1. - Vaclav Kotesovec, Jul 04 2025