cp's OEIS Frontend

A360465 E.g.f. satisfies A(x) = exp(x * exp(2*x) * A(x)).

%I A360465 #17 Feb 16 2025 08:34:04
%S A360465 1,1,7,64,829,14056,295399,7426252,217637305,7291538704,275050426411,
%T A360465 11540336658676,533224609095061,26908386824872216,1472691380336896399,
%U A360465 86892807951798473116,5498668489586321670769,371511527654280649783840
%N A360465 E.g.f. satisfies A(x) = exp(x * exp(2*x) * A(x)).
%H A360465 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A360465 E.g.f.: A(x) = exp( -LambertW(-x * exp(2*x)) ).
%F A360465 E.g.f.: A(x) = -LambertW(-x * exp(2*x)) / (x * exp(2*x)).
%F A360465 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x * exp(2*x))^k / k!.
%F A360465 a(n) = Sum_{k=0..n} (2*k)^(n-k) * (k+1)^(k-1) * binomial(n,k).
%F A360465 a(n) ~ sqrt(1+LambertW(2*exp(-1))) * 2^n * n^(n-1) / (exp(n-1) * LambertW(2*exp(-1))^n). - _Vaclav Kotesovec_, Feb 08 2023
%o A360465 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*exp(2*x)))))
%o A360465 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-lambertw(-x*exp(2*x))/(x*exp(2*x))))
%o A360465 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x*exp(2*x))^k/k!)))
%o A360465 (PARI) a(n) = sum(k=0, n, (2*k)^(n-k)*(k+1)^(k-1)*binomial(n, k));
%Y A360465 Cf. A038050, A273954, A357247, A360466.
%K A360465 nonn
%O A360465 0,3
%A A360465 _Seiichi Manyama_, Feb 08 2023