A129380 Partial sums of A129379.

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

Offset

1,2

Links

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

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A000217, A052560, A129379.

Sequence in context: A136121 A023369 A358058 * A085866 A282231 A301522

Adjacent sequences: A129377 A129378 A129379 * A129381 A129382 A129383

Keyword

nonn

Author

Reinhard Zumkeller, Apr 14 2007

Status

approved