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a(n) = Sum_{k=0..n} binomial(n+k+2,k)*binomial(2*n+1,n-k).
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%I #42 Apr 10 2017 22:56:54

%S 1,7,50,364,2688,20064,151008,1144000,8712704,66646528,511673344,

%T 3940579328,30429184000,235521884160,1826663915520,14192851599360,

%U 110453212446720,860819570688000,6717522904350720,52482715893104640

%N a(n) = Sum_{k=0..n} binomial(n+k+2,k)*binomial(2*n+1,n-k).

%H Seiichi Manyama, <a href="/A275827/b275827.txt">Table of n, a(n) for n = 0..1108</a>

%F G.f.: -(sqrt(1-8*x)+3)/(sqrt(1-8*x)*(8*x-2)+16*x-2).

%F a(n) ~ 3*8^n/sqrt(Pi*n). - _Ilya Gutkovskiy_, Nov 24 2016

%t f[n_] := Sum[ Binomial[n + k + 2, k] Binomial[2n + 1, n - k], {k, 0, n}]; Array[f, 21, 0] (* or *)

%t CoefficientList[ Series[(1 - Sqrt[1 - 8x] - 2 x - 2Sqrt[1 - 8x] x)/(16Sqrt[1 - 8x] x^2), {x, 0, 20}], x] (* _Robert G. Wilson v_, Nov 23 2016 *)

%o (Maxima)

%o taylor(-(sqrt(1-8*x)+3)/(sqrt(1-8*x)*(8*x-2)+16*x-2),x,0,20);

%o (PARI) x='x+O('x^50); Vec(-(sqrt(1-8*x)+3)/(sqrt(1-8*x)*(8*x-2)+16*x-2)) \\ _G. C. Greubel_, Apr 10 2017

%Y Cf. A000108.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Nov 23 2016