A130031 Row sums of triangle A129467.

1, 1, -1, 5, -55, 1045, -30305, 1242505, -68337775, 4851982025, -431826400225, 47069077624525, -6166049168812775, 955737621165980125, -172988509431042402625, 36154598471087862148625, -8640949034589999053521375, 2341697188373889743504292625

Offset: 0

Author

Wolfdieter Lang, May 04 2007

Keywords

• sign
• easy

Comments

See the M. Bruschi et al. reference given in A129467.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A129467, A130559.

Programs

Magma

[1] cat [n le 1 select 1 else -(n^2-n-1)*Self(n-1): n in [1..30]]; // G. C. Greubel, Feb 10 2024

Mathematica

Table[(-1)^n*Product[k^2-k-1, {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 24 2016 *)
Table[FullSimplify[(-1)^n * Cos[Sqrt[5]*Pi/2] * Gamma[n+(Sqrt[5]+1)/2] * Gamma[n-(Sqrt[5]-1)/2]/Pi], {n, 0, 20}] (* Vaclav Kotesovec, Aug 24 2016 *)

SageMath

def A130031(n): return 1 if n<2 else -(n^2-n-1)*A130031(n-1)
[A130031(n) for n in range(31)] # G. C. Greubel, Feb 10 2024

Formula

a(n) = Sum_{j=0..n} A129467(n,j), n >= 0.

a(n) = Sum_{j=0..n-1} A130559(n, j), n>= 1.

From Vaclav Kotesovec, Aug 24 2016: (Start)

a(n) = (-1)^n*Product_{k=1..n} (k^2 - k - 1).

a(n) ~ 2*(-1)^n * cos(sqrt(5)*Pi/2) * n^(2*n) / exp(2*n). (End)

a(n) + (n^2-n-1)*a(n-1) = 0. - R. J. Mathar, Jan 21 2018