A362314 - OEIS

%I #17 Apr 18 2023 10:49:29

%S 1,1,1,1,25,151,541,1471,84001,925345,5682601,25177681,2245355641,

%T 35901100951,312222474565,1917363070351,232479594721921,

%U 4873115730725761,54830346428307601,430468886732009185,65997947903313461401,1711564302775814535511

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

%H Winston de Greef, <a href="/A362314/b362314.txt">Table of n, a(n) for n = 0..431</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^4/4).

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

%F From _Vaclav Kotesovec_, Apr 18 2023: (Start)

%F a(n) ~ c * n^n / exp(3*n/4), where

%F c = cosh(1) + cos(1) if mod(n,4)=0,

%F c = sinh(1) + sin(1) if mod(n,4)=1,

%F c = cosh(1) - cos(1) if mod(n,4)=2,

%F c = sinh(1) - sin(1) if mod(n,4)=3. (End)

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

%Y Cf. A277614, A362300, A362319.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Apr 15 2023