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A362314 a(n) = n! * Sum_{k=0..floor(n/4)} (n/4)^k /(k! * (n-4*k)!).

%I A362314 #18 Feb 16 2025 08:34:05
%S A362314 1,1,1,1,25,151,541,1471,84001,925345,5682601,25177681,2245355641,
%T A362314 35901100951,312222474565,1917363070351,232479594721921,
%U A362314 4873115730725761,54830346428307601,430468886732009185,65997947903313461401,1711564302775814535511
%N A362314 a(n) = n! * Sum_{k=0..floor(n/4)} (n/4)^k /(k! * (n-4*k)!).
%H A362314 Winston de Greef, <a href="/A362314/b362314.txt">Table of n, a(n) for n = 0..431</a>
%H A362314 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362314 a(n) = n! * [x^n] exp(x + n*x^4/4).
%F A362314 E.g.f.: exp( ( -LambertW(-x^4) )^(1/4) ) / (1 + LambertW(-x^4)).
%F A362314 From _Vaclav Kotesovec_, Apr 18 2023: (Start)
%F A362314 a(n) ~ c * n^n / exp(3*n/4), where
%F A362314 c = cosh(1) + cos(1) if mod(n,4)=0,
%F A362314 c = sinh(1) + sin(1) if mod(n,4)=1,
%F A362314 c = cosh(1) - cos(1) if mod(n,4)=2,
%F A362314 c = sinh(1) - sin(1) if mod(n,4)=3. (End)
%o A362314 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-x^4))^(1/4))/(1+lambertw(-x^4))))
%Y A362314 Cf. A277614, A362300, A362319.
%K A362314 nonn
%O A362314 0,5
%A A362314 _Seiichi Manyama_, Apr 15 2023