A356796 - OEIS

%I #23 Sep 03 2022 09:55:06

%S 1,0,2,3,92,450,14454,141540,4980128,78711696,3048567480,68677353360,

%T 2930551701384,86832573553440,4079649847428960,150444517302424800,

%U 7768028697749806080,342721736137376184960,19392702029822685015360,994397473912386435004800

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

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

%F a(n) = n! * Sum_{k=0..floor(n/2)} (3*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

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

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

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

%o (PARI) a(n) = n!*sum(k=0, n\2, (3*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*(-x*log(1-x))^k/k!)))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*log(1-x))/3)))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*log(1-x))/(3*x*log(1-x)))^(1/3)))

%Y Cf. A066166, A355842, A356795.

%Y Cf. A356787.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 28 2022