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a(n) = binomial(n+3,3)*binomial(n+8,3).
2

%I #14 Jan 06 2023 19:39:29

%S 56,336,1200,3300,7700,16016,30576,54600,92400,149600,233376,352716,

%T 518700,744800,1047200,1445136,1961256,2622000,3458000,4504500,

%U 5801796,7395696,9338000,11687000,14508000,17873856,21865536,26572700,32094300,38539200,46026816

%N a(n) = binomial(n+3,3)*binomial(n+8,3).

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F From _R. J. Mathar_, Nov 29 2015: (Start)

%F a(n) = A000292(n+1)*A000292(n+6) = 4*A033276(n+6).

%F G.f.: 4*(-14+14*x-6*x^2+x^3)/(x-1)^7. (End)

%F From _Amiram Eldar_, Aug 30 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 109/4900.

%F Sum_{n>=0} (-1)^n/a(n) = 48*log(2)/35 - 2291/2450. (End)

%e If n=0 then C(0+3,0+0)*C(0+8,3) = C(3,0)*C(8,3) = 1*56 = 56.

%e If n=8 then C(8+3,8+0)*C(8+8,3) = C(11,8)*C(16,3) = 165*560 = 92400.

%t a[n_] := Binomial[n+3, 3] * Binomial[n+8, 3]; Array[a, 30, 0] (* _Amiram Eldar_, Aug 30 2022 *)

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{56,336,1200,3300,7700,16016,30576},40] (* _Harvey P. Dale_, Jan 06 2023 *)

%Y Cf. A000292, A033276, A062190.

%K easy,nonn

%O 0,1

%A _Zerinvary Lajos_, Apr 22 2005