A371524 - OEIS

%I #41 Feb 16 2025 08:34:06

%S 1,4,20,124,936,8424,88648,1072432,14702720,225692128,3839770656,

%T 71780577312,1463532416320,32337850727680,770039603953664,

%U 19664621381714944,536234348295180288,15554459021934423552,478297493455731968512,15543431292269887979008

%N E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + A(x)^(1/4)) ).

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

%F E.g.f.: A(x) = exp( 2*x - 4*LambertW(-x/2 * exp(x/2)) ).

%F If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).

%F G.f.: 2 * Sum_{k>=0} (k/2+2)^(k-1) * x^k/(1 - (k/2+2)*x)^(k+1).

%F a(n) ~ sqrt(LambertW(exp(-1)) + 1) * n^(n-1) / (2^(n-2) * exp(n) * LambertW(exp(-1))^(n+4)). - _Vaclav Kotesovec_, Apr 24 2024

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x-4*lambertw(-x/2*exp(x/2)))))

%o (PARI) a(n, r=2, t=0, u=1/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));

%o (PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (k/2+2)^(k-1)*x^k/(1-(k/2+2)*x)^(k+1)))

%Y Cf. A007889, A372236.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 23 2024