A356795 - OEIS

%I #21 Feb 16 2025 08:34:03

%S 1,0,2,3,68,330,7674,73080,1883440,28281960,818625960,17120406600,

%T 557507325000,15014517495120,548643259812816,18056683281775320,

%U 736892260092195840,28579282973977498560,1295028345251832359616,57666859088090317591680

%N E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)^2).

%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/2)} (2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

%F E.g.f.: A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (-x * log(1-x))^k / k!.

%F E.g.f.: A(x) = exp( -LambertW(2 * x * log(1-x))/2 ).

%F E.g.f.: A(x) = ( LambertW(2 * x * log(1-x))/(2 * x * log(1-x)) )^(1/2).

%o (PARI) a(n) = n!*sum(k=0, n\2, (2*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, (2*k+1)^(k-1)*(-x*log(1-x))^k/k!)))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*log(1-x))/2)))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*log(1-x))/(2*x*log(1-x)))^(1/2)))

%Y Cf. A066166, A355842, A356796.

%Y Cf. A356786.

%K nonn,changed

%O 0,3

%A _Seiichi Manyama_, Aug 28 2022