A360596 - OEIS

%I #16 Feb 13 2023 11:51:49

%S 1,3,22,282,5224,126120,3742704,131612432,5347866752,246490091136,

%T 12704900911360,724072211436288,45209213973292032,3068872654856532992,

%U 225023336997933996032,17724257054969009940480,1492513932494133333753856,133800772458366199028023296

%N Expansion of e.g.f. 1/( (1 - x) * (1 + LambertW(-2*x)) ).

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

%F a(n) = n! * Sum_{k=0..n} (2*k)^k / k!.

%F a(0)=1; a(n) = n*a(n-1) + (2*n)^n.

%F a(n) ~ 2^(n+1) * n^n / (2 - exp(-1)). - _Vaclav Kotesovec_, Feb 13 2023

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

%o (PARI) a(n) = n!*sum(k=0, n, (2*k)^k/k!);

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+(2*i)^i); v;

%Y Cf. A062971, A277506, A277509.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Feb 13 2023