cp's OEIS Frontend

A356949 E.g.f. satisfies log(A(x)) = x^2 * (exp(x) - 1) * A(x).

%I A356949 #17 Feb 16 2025 08:34:03
%S A356949 1,0,0,6,12,20,1110,7602,35336,1103832,14984010,134552990,3457329612,
%T A356949 70828191876,1017237973934,25648737955050,676111332667920,
%U A356949 13760810592066992,373071111301807506,11594147432172228918,307097278689726728660,9330499711181779575900
%N A356949 E.g.f. satisfies log(A(x)) = x^2 * (exp(x) - 1) * A(x).
%H A356949 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356949 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * Stirling2(n-2*k,k)/(n-2*k)!.
%F A356949 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^2 * (exp(x) - 1))^k / k!.
%F A356949 E.g.f.: A(x) = exp( -LambertW(x^2 * (1 - exp(x))) ).
%F A356949 E.g.f.: A(x) = LambertW(x^2 * (1 - exp(x)))/(x^2 * (1 - exp(x))).
%t A356949 nmax = 21; A[_] = 1;
%t A356949 Do[A[x_] = Exp[(-1 + Exp[x])*A[x]*x^2] + O[x]^(nmax+1) // Normal, {nmax}];
%t A356949 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356949 (PARI) a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!);
%o A356949 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^2*(exp(x)-1))^k/k!)))
%o A356949 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2*(1-exp(x))))))
%o A356949 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2*(1-exp(x)))/(x^2*(1-exp(x)))))
%Y A356949 Cf. A052880, A355843, A356950.
%Y A356949 Cf. A000272, A240989, A355287, A356797, A356951.
%K A356949 nonn
%O A356949 0,4
%A A356949 _Seiichi Manyama_, Sep 06 2022