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a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) / (k! * (n-3*k)!).
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%I #14 May 03 2023 09:09:57

%S 1,1,1,7,25,61,1201,7771,30577,1058905,9904321,53722351,2708688841,

%T 33126146197,228967340785,15262865820931,230517745701601,

%U 1936173471789361,161021598306402817,2894434429492525015,28614958982310290041

%N a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) / (k! * (n-3*k)!).

%H Seiichi Manyama, <a href="/A362523/b362523.txt">Table of n, a(n) for n = 0..427</a>

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

%F E.g.f.: exp(x - LambertW(-x^3)) = -LambertW(-x^3)/x^3 * exp(x).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3))))

%Y Cf. A088957, A362522.

%Y Cf. A089464, A362348.

%K nonn,easy

%O 0,4

%A _Seiichi Manyama_, Apr 23 2023