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%I #18 Mar 04 2024 08:16:19
%S 1,0,0,0,4,10,40,210,784,5040,40200,369600,5285280,72072000,
%T 1006889520,14760345600,210510263040,3131345817600,49229619129600,
%U 818940523564800,15054020163619200,301204611031564800,6455999452413772800,146587705490513548800
%N E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3/6).
%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/4)} (-k+1)^(k-1) * |Stirling1(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 * log(1-x))^k / k!.
%F E.g.f.: A(x) = exp( LambertW(-x^3/6 * log(1-x)) ).
%F E.g.f.: A(x) = -x^3/6 * log(1-x)/LambertW(-x^3/6 * log(1-x)).
%t nmax = 23; A[_] = 1;
%t Do[A[x_] = ((1 - x)^(-x^3/6))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o (PARI) a(n) = n!*sum(k=0, n\4, (-k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(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*log(1-x))^k/k!)))
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3/6*log(1-x)))))
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3/6*log(1-x)/lambertw(-x^3/6*log(1-x))))
%Y Cf. A356905, A356912.
%Y Cf. A351493, A356753.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Sep 03 2022
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