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%I #23 Feb 04 2024 21:20:03
%S 1,3,6,12,48,336,3696,59136,1300992,37728768,1395964416,64214363136,
%T 3596004335616,240932290486272,19033650948415488,1751095887254224896,
%U 185616164048947838976,22459555849922688516096
%N Partial sums of A129379.
%H G. C. Greubel, <a href="/A129380/b129380.txt">Table of n, a(n) for n = 1..270</a>
%F a(n) = A129379(n+1)/A000217(n-2) for n>2.
%F a(n) = a(n-1) + A129379(n) for n > 1, a(1) = 1.
%F From _G. C. Greubel_, Feb 03 2024: (Start)
%F a(n) = (6/2^(n-3))*|Pochhammer((3+i*sqrt(7))/2, n-3)|^2, for n > 2.
%F a(n) = (3/2^(n-3))*Product_{k=0..n-2} (k^2 - k + 2), for n > 2.
%F a(n) = (1/2)*(n^2 - 5*n + 8)*a(n-1), with a(1) = 1, a(2) = 3, a(3) = 6. (End)
%F From _Vaclav Kotesovec_, Feb 03 2024: (Start)
%F For n>=3, a(n) = 3 * cosh(sqrt(7)*Pi/2) * 2^(3-n) * Gamma(n - 3/2 - i*sqrt(7)/2) * Gamma(n - 3/2 + i*sqrt(7)/2)/Pi, where i is the imaginary unit.
%F a(n) ~ 3 * cosh(sqrt(7)*Pi/2) * n^(2*n-4) / (2^(n-4) * exp(2*n)). (End)
%t a[n_]:= a[n]= If[n<4, Binomial[n+1,2], (n^2-5*n+8)*a[n-1]/2];
%t Table[a[n], {n,40}] (* _G. C. Greubel_, Feb 03 2024 *)
%t Round[Flatten[{{1, 3}, Table[(3*2^(3-n) * Cosh[Sqrt[7]*Pi/2] * Gamma[n - 3/2 - I*Sqrt[7]/2] * Gamma[n - 3/2 + I*Sqrt[7]/2])/Pi, {n, 3, 20}]}]] (* _Vaclav Kotesovec_, Feb 03 2024 *)
%o (Magma)
%o A129380:= func< n | n le 2 select 2*n-1 else (3/2^(n-2))*(&*[k^2-k+2: k in [0..n-2]]) >;
%o [A129380(n): n in [1..40]]; // _G. C. Greubel_, Feb 03 2024
%o (SageMath)
%o def A129380(n): return 2*n-1 if n<3 else 3*product(j^2-j+2 for j in range(n-1))//2^(n-2)
%o [A129380(n) for n in range(1, 41)] # _G. C. Greubel_, Feb 03 2024
%Y Cf. A000217, A052560, A129379.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Apr 14 2007
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