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a(n) = [x^n] (1/6 * Sum_{k=0..n} (k+3)!*x^k)^(1/2).
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%I #13 Apr 27 2018 10:32:10

%S 1,2,8,44,300,2408,22056,225824,2547352,31322640,416442016,5948450912,

%T 90815006528,1475513165184,25419300628160,462839142264960,

%U 8882065110092000,179190173518717120,3791700201187720960,83977684252758211200,1942984028971973710720

%N a(n) = [x^n] (1/6 * Sum_{k=0..n} (k+3)!*x^k)^(1/2).

%H Seiichi Manyama, <a href="/A303613/b303613.txt">Table of n, a(n) for n = 0..447</a>

%o (PARI) N=66; x='x+O('x^N); Vec((1/3!*sum(k=0, N, (k+3)!*x^k))^(1/2))

%Y Cf. A001715, A303568.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 27 2018