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a(n) = n! * Sum_{k=0..floor(n/4)} (n/24)^k /(k! * (n-4*k)!).
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%I #19 Apr 18 2023 08:28:46

%S 1,1,1,1,5,26,91,246,2801,26650,159601,702406,12479941,172561676,

%T 1462655195,8918930476,215370384321,3906667179836,42828875064001,

%U 333816101642140,10190496077676901,228789539391769336,3077152545301687931,29203537040556576776

%N a(n) = n! * Sum_{k=0..floor(n/4)} (n/24)^k /(k! * (n-4*k)!).

%H Seiichi Manyama, <a href="/A362317/b362317.txt">Table of n, a(n) for n = 0..465</a>

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

%F a(n) = n! * [x^n] exp(x + n*x^4/24).

%F E.g.f.: exp( ( -6*LambertW(-x^4/6) )^(1/4) ) / (1 + LambertW(-x^4/6)).

%t nmax = 30; CoefficientList[PowerExpand[Series[E^((-6*LambertW[-x^4/6])^(1/4)) / (1 + LambertW[-x^4/6]), {x, 0, nmax}]], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Apr 18 2023 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-6*lambertw(-x^4/6))^(1/4))/(1+lambertw(-x^4/6))))

%Y Cf. A362173, A362336.

%Y Cf. A351932, A362314.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Apr 16 2023