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A355287 E.g.f. satisfies A(x) = 1/(1 - x)^(x^2 * A(x)).

%I A355287 #21 Feb 16 2025 08:34:03
%S A355287 1,0,0,6,12,40,1260,8568,62160,1473120,19111680,232626240,5403451680,
%T A355287 103176028800,1822033204992,45916616592000,1129459815993600,
%U A355287 26346457488798720,749439127417466880,22165051763204582400,640916967497214643200,20787453048015928350720
%N A355287 E.g.f. satisfies A(x) = 1/(1 - x)^(x^2 * A(x)).
%H A355287 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A355287 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.
%F A355287 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^2 * log(1-x))^k / k!.
%F A355287 E.g.f.: A(x) = exp( -LambertW(x^2 * log(1-x)) ).
%F A355287 E.g.f.: A(x) = LambertW(x^2 * log(1-x))/(x^2 * log(1-x)).
%t A355287 nmax = 21; A[_] = 1;
%t A355287 Do[A[x_] = 1/(1 - x)^(x^2*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A355287 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A355287 (PARI) a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);
%o A355287 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^2*log(1-x))^k/k!)))
%o A355287 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2*log(1-x)))))
%o A355287 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2*log(1-x))/(x^2*log(1-x))))
%Y A355287 Cf. A353228, A356910.
%K A355287 nonn
%O A355287 0,4
%A A355287 _Seiichi Manyama_, Sep 03 2022