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A360748 Expansion of Sum_{k>=0} (x * (1 + k*x^2))^k.

%I A360748 #18 Feb 20 2023 07:12:22
%S A360748 1,1,1,2,5,10,21,53,133,327,861,2361,6469,18168,52757,155221,463077,
%T A360748 1412656,4379917,13747504,43834213,141866555,464650309,1541008295,
%U A360748 5176660997,17586913779,60400627453,209746820056,735953607173,2607716976945,9330605338485
%N A360748 Expansion of Sum_{k>=0} (x * (1 + k*x^2))^k.
%H A360748 Vaclav Kotesovec, <a href="/A360748/a360748.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A360748 a(n) = Sum_{k=0..floor(n/3)} (n-2*k)^k * binomial(n-2*k,k).
%F A360748 a(n) ~ exp(exp(2/3)*n^(2/3)/3^(2/3) - 5*exp(4/3)*n^(1/3)/(18*3^(1/3)) + 22*exp(2)/81) * n^(n/3) / 3^(n/3 + 1) * (1 + (2*exp(2/3)/3^(5/3) - 3295*exp(8/3)/(2916*3^(2/3)))/n^(1/3) + (3^(2/3)/(8*exp(2/3)) + 35*exp(4/3)/(36*3^(1/3)) + 27379*exp(10/3)/(17496*3^(1/3)) + 10857025*exp(16/3)/(51018336*3^(1/3)))/n^(2/3)). - _Vaclav Kotesovec_, Feb 20 2023
%t A360748 Join[{1},Table[Sum[Binomial[n - 2*k,k] * (n - 2*k)^k, {k,0,n/3}], {n,1,30}]] (* _Vaclav Kotesovec_, Feb 20 2023 *)
%o A360748 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+k*x^2))^k))
%o A360748 (PARI) a(n) = sum(k=0, n\3, (n-2*k)^k*binomial(n-2*k, k));
%Y A360748 Cf. A360592, A360749.
%Y A360748 Cf. A000930, A360730, A360479.
%K A360748 nonn
%O A360748 0,4
%A A360748 _Seiichi Manyama_, Feb 19 2023