A356952 - OEIS

%I #17 Feb 16 2025 08:34:04

%S 1,0,0,0,4,10,20,35,1736,15204,88320,415965,7632460,121801966,

%T 1368227224,12184672955,176889193040,3490851044360,59703361471296,

%U 837948141904569,13407228541467540,283596013866706450,6226093732482731800,121326684752194084471

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

%F E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^3/6 * (exp(x) - 1))^k / k!.

%F E.g.f.: A(x) = exp( -LambertW(x^3/6 * (1 - exp(x))) ).

%F E.g.f.: A(x) = LambertW(x^3/6 * (1 - exp(x)))/(x^3/6 * (1 - exp(x))).

%t nmax = 23; A[_] = 1;

%t Do[A[x_] = Exp[x^3/6*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)

%o (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 (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 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3/6*(1-exp(x))))))

%o (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 Cf. A052880, A355843, A356951.

%Y Cf. A000272, A354001, A356753, A356950.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Sep 06 2022