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Expansion of e.g.f. -LambertW(x^3 * log(1-x))/6.
1

%I #16 Sep 24 2022 08:15:01

%S 0,0,0,0,4,10,40,210,8064,70560,640800,6375600,189383040,3165402240,

%T 48879754560,762766804800,21652937349120,525738717504000,

%U 11796584629939200,259139188966694400,7842638783736115200,240231375437935795200,7066934411387842252800

%N Expansion of e.g.f. -LambertW(x^3 * log(1-x))/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!/6) * Sum_{k=1..floor(n/4)} k^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.

%t With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^3 * Log[1 - x]]/6, {x, 0, m}], x]] (* _Amiram Eldar_, Sep 24 2022 *)

%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3*log(1-x)))/6))

%o (PARI) a(n) = n!*sum(k=1, n\4, k^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!)/6;

%Y Cf. A052807, A355874, A357265.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Sep 24 2022