This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381415 #12 Jul 04 2025 05:24:04
%S A381415 1,1,5,50,765,15852,415441,13182976,491502521,21061603152,
%T A381415 1020066862269,55107133707232,3285531022228725,214295961023511616,
%U A381415 15179005200468020489,1160334809344169734144,95214513195493336071537,8347897781857074205573376,778804910740650550851809013
%N A381415 E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)^2) ).
%F A381415 a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * A136630(n,k).
%F A381415 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
%t A381415 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 *)
%o A381415 (PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
%o A381415 a(n) = sum(k=0, n, (2*n+1)^(k-1)*a136630(n, k));
%Y A381415 Cf. A136630, A162650.
%K A381415 nonn
%O A381415 0,3
%A A381415 _Seiichi Manyama_, Feb 23 2025