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

%I #14 Jan 09 2024 08:46:33

%S 1,0,1,2,19,100,1118,10034,134993,1715140,27589661,449763360,

%T 8522965956,168431719308,3698624353289,85523954588806,

%U 2142927489388319,56618555339223572,1596938935380604858,47399670488829289678,1487559109670284821841

%N Expansion of Sum_{k>=0} (k*x)^(2*k) / (1 - k*x)^(k+1).

%H Seiichi Manyama, <a href="/A360816/b360816.txt">Table of n, a(n) for n = 0..418</a>

%F a(n) = Sum_{k=0..floor(n/2)} k^n * binomial(n-k,k).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^(2*k)/(1-k*x)^(k+1)))

%o (PARI) a(n) = sum(k=0, n\2, k^n*binomial(n-k, k));

%Y Cf. A072034, A360817.

%Y Cf. A360814.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Feb 21 2023