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A267522 a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3.

%I A267522 #34 Feb 23 2024 20:13:41
%S A267522 8,56,176,400,760,1288,2016,2976,4200,5720,7568,9776,12376,15400,
%T A267522 18880,22848,27336,32376,38000,44240,51128,58696,66976,76000,85800,
%U A267522 96408,107856,120176,133400,147560,162688,178816,195976,214200,233520,253968,275576,298376,322400
%N A267522 a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3.
%C A267522 Partial sums of A152750.
%H A267522 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A267522 G.f.: 8*(1 + 3*x)/(1 - x)^4.
%F A267522 E.g.f.: (4/3)*exp(x)*(6 + 36*x + 27*x^2 + 4*x^3).
%F A267522 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A267522 a(n) = 4*A268684(n + 1).
%F A267522 Sum_{n>=0} 1/a(n) = -3*(2*Pi - 12*log(2) + 1)/20 = 0.15518712893...
%F A267522 a(n) = 8*A002412(n+1). - _Yasser Arath Chavez Reyes_, Feb 23 2024
%e A267522 a(0) = (0 + 2)*(1 + 3) = 8;
%e A267522 a(1) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) = 56;
%e A267522 a(2) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) = 176;
%e A267522 a(3) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) + (6 + 8)*(7 + 9) = 400, etc
%t A267522 Table[(4 (n + 1)) (n + 2) ((4 n + 3)/3), {n, 0, 38}]
%t A267522 LinearRecurrence[{4, -6, 4, -1}, {8, 56, 176, 400}, 39]
%o A267522 (PARI) a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3; \\ _Michel Marcus_, Apr 10 2016
%o A267522 (PARI) x='x+O('x^99); Vec(8*(1+3*x)/(1-x)^4) \\ _Altug Alkan_, Apr 10 2016
%Y A267522 Cf. A002412, A152750, A268684.
%K A267522 nonn,easy
%O A267522 0,1
%A A267522 _Ilya Gutkovskiy_, Apr 09 2016