|
%I #17 Feb 18 2023 19:00:40
%S 1,1,4,28,264,3206,47684,839249,17058688,393216567,10134918592,
%T 288815780665,9016571143680,306027510946208,11219450971161024,
%U 441846991480590475,18602901833071633792,833832341625621777368,39642569136740054367808
%N Expansion of Sum_{k>=0} (k * x * (1 + x^2))^k.
%H Winston de Greef, <a href="/A360727/b360727.txt">Table of n, a(n) for n = 0..385</a>
%F a(n) = Sum_{k=0..floor(n/3)} (n-2*k)^(n-2*k) * binomial(n-2*k,k).
%F a(n) ~ n^n * (1 + exp(-2)/n + exp(-4)/(2*n^2)). - _Vaclav Kotesovec_, Feb 18 2023
%t nmax = 20; CoefficientList[1 + Series[Sum[(k*x*(1 + x^2))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 18 2023 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+x^2))^k))
%o (PARI) a(n) = sum(k=0, n\3, (n-2*k)^(n-2*k)*binomial(n-2*k, k));
%Y Cf. A360611, A360728.
%Y Cf. A360730.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Feb 18 2023
|
