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 A372162 #16 Feb 16 2025 08:34:06
%S A372162 1,1,5,31,329,4201,70357,1374703,31888817,839198737,25021698821,
%T A372162 827967913279,30240609486265,1205630521463161,52177446181578005,
%U A372162 2434309587346377871,121857094322821338593,6513265883385904609057,370302655720337288548741
%N A372162 E.g.f. A(x) satisfies A(x) = exp( x * sqrt(1+2*x) * A(x) ).
%H A372162 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A372162 E.g.f.: A(x) = exp( -LambertW(-x * sqrt(1+2*x)) ).
%F A372162 a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * 2^(n-k) * binomial(k/2,n-k)/k!.
%F A372162 From _Vaclav Kotesovec_, Apr 21 2024: (Start)
%F A372162 E.g.f.: -LambertW(-x*sqrt(1 + 2*x))/(x*sqrt(1 + 2*x)).
%F A372162 a(n) ~ sqrt(3*r + 1) * n^(n-1) / ((1 + 2*r)^(3/4) * exp(n - 1/2) * r^(n + 1/2)), where r = (exp(2/3) + (-exp(1) + (6*(9 + sqrt(81 - 3*exp(2))))/exp(1))^(2/3)) / (6*(54 - exp(2) + 6*sqrt(81 - 3*exp(2)))^(1/3)) - 1/6 = 0.292252770550601628... (End)
%o A372162 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*sqrt(1+2*x)))))
%o A372162 (PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*2^(n-k)*binomial(k/2, n-k)/k!);
%Y A372162 Cf. A362771, A372163.
%K A372162 nonn
%O A372162 0,3
%A A372162 _Seiichi Manyama_, Apr 20 2024