A129380 Partial sums of A129379.

1, 3, 6, 12, 48, 336, 3696, 59136, 1300992, 37728768, 1395964416, 64214363136, 3596004335616, 240932290486272, 19033650948415488, 1751095887254224896, 185616164048947838976, 22459555849922688516096

Offset: 1

Author

Reinhard Zumkeller, Apr 14 2007

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..270

Crossrefs

Cf. A000217, A052560, A129379.

Programs

Magma

A129380:= func< n | n le 2 select 2*n-1 else (3/2^(n-2))*(&*[k^2-k+2: k in [0..n-2]])  >;
[A129380(n): n in [1..40]]; // G. C. Greubel, Feb 03 2024

Mathematica

a[n_]:= a[n]= If[n<4, Binomial[n+1,2], (n^2-5*n+8)*a[n-1]/2];
Table[a[n], {n,40}] (* G. C. Greubel, Feb 03 2024 *)
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 *)

SageMath

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)
[A129380(n) for n in range(1, 41)] # G. C. Greubel, Feb 03 2024

Formula

a(n) = A129379(n+1)/A000217(n-2) for n>2.

a(n) = a(n-1) + A129379(n) for n > 1, a(1) = 1.

From G. C. Greubel, Feb 03 2024: (Start)

a(n) = (6/2^(n-3))*|Pochhammer((3+i*sqrt(7))/2, n-3)|^2, for n > 2.

a(n) = (3/2^(n-3))*Product_{k=0..n-2} (k^2 - k + 2), for n > 2.

a(n) = (1/2)*(n^2 - 5*n + 8)*a(n-1), with a(1) = 1, a(2) = 3, a(3) = 6. (End)

From Vaclav Kotesovec, Feb 03 2024: (Start)

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.

a(n) ~ 3 * cosh(sqrt(7)*Pi/2) * n^(2*n-4) / (2^(n-4) * exp(2*n)). (End)