cp's OEIS Frontend

A356952 E.g.f. satisfies log(A(x)) = x^3/6 * (exp(x) - 1) * A(x).

%I A356952 #17 Feb 16 2025 08:34:04
%S A356952 1,0,0,0,4,10,20,35,1736,15204,88320,415965,7632460,121801966,
%T A356952 1368227224,12184672955,176889193040,3490851044360,59703361471296,
%U A356952 837948141904569,13407228541467540,283596013866706450,6226093732482731800,121326684752194084471
%N A356952 E.g.f. satisfies log(A(x)) = x^3/6 * (exp(x) - 1) * A(x).
%H A356952 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356952 a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).
%F A356952 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^3/6 * (exp(x) - 1))^k / k!.
%F A356952 E.g.f.: A(x) = exp( -LambertW(x^3/6 * (1 - exp(x))) ).
%F A356952 E.g.f.: A(x) = LambertW(x^3/6 * (1 - exp(x)))/(x^3/6 * (1 - exp(x))).
%t A356952 nmax = 23; A[_] = 1;
%t A356952 Do[A[x_] = Exp[x^3/6*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t A356952 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A356952 (PARI) a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));
%o A356952 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^3/6*(exp(x)-1))^k/k!)))
%o A356952 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3/6*(1-exp(x))))))
%o A356952 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3/6*(1-exp(x)))/(x^3/6*(1-exp(x)))))
%Y A356952 Cf. A052880, A355843, A356951.
%Y A356952 Cf. A000272, A354001, A356753, A356950.
%K A356952 nonn
%O A356952 0,5
%A A356952 _Seiichi Manyama_, Sep 06 2022