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 A386366 #13 Jul 19 2025 18:14:02
%S A386366 1,1,11,225,6729,264885,12933675,753953445,51089936625,3945857018985,
%T A386366 342128949720075,32905744117871625,3476617058554464825,
%U A386366 400259518407091468125,49874289081145099245675,6687208401827555535058125,960003161392360306947350625,146914452707464363053984476625
%N A386366 Expansion of e.g.f. (Sum_{k>=0} binomial(4*k,k) * x^k)^(1/4).
%C A386366 In general, if m > 1 and e.g.f. = (Sum_{k>=0} binomial(m*k,k) * x^k)^(1/m), then a(n) ~ n! * m^(m*n + 1/(2*m)) / (Gamma(1/(2*m)) * 2^(1/(2*m)) * n^(1 - 1/(2*m)) * (m-1)^((m-1)*n + 1/(2*m))). - _Vaclav Kotesovec_, Jul 19 2025
%F A386366 a(n) ~ sqrt(Pi) * 2^(8*n + 5/8) * n^(n - 3/8) / (Gamma(1/8) * exp(n) * 3^(3*n + 1/8)). - _Vaclav Kotesovec_, Jul 19 2025
%t A386366 nmax = 20; CoefficientList[Series[Sum[Binomial[4*k,k] * x^k, {k, 0, nmax}]^(1/4), {x, 0, nmax}], x] * Range[0,nmax]! (* _Vaclav Kotesovec_, Jul 19 2025 *)
%o A386366 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(4*k, k)*x^k)^(1/4)))
%Y A386366 Cf. A000142, A007696, A386365.
%Y A386366 Cf. A378484.
%K A386366 nonn
%O A386366 0,3
%A A386366 _Seiichi Manyama_, Jul 19 2025