A130031 - OEIS

%I #14 Feb 10 2024 02:59:54

%S 1,1,-1,5,-55,1045,-30305,1242505,-68337775,4851982025,-431826400225,

%T 47069077624525,-6166049168812775,955737621165980125,

%U -172988509431042402625,36154598471087862148625,-8640949034589999053521375,2341697188373889743504292625

%N Row sums of triangle A129467.

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

%H G. C. Greubel, <a href="/A130031/b130031.txt">Table of n, a(n) for n = 0..250</a>

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

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

%F From _Vaclav Kotesovec_, Aug 24 2016: (Start)

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

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

%F a(n) + (n^2-n-1)*a(n-1) = 0. - _R. J. Mathar_, Jan 21 2018

%t Table[(-1)^n*Product[k^2-k-1, {k, 1, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 24 2016 *)

%t 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 *)

%o (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

%o (SageMath)

%o def A130031(n): return 1 if n<2 else -(n^2-n-1)*A130031(n-1)

%o [A130031(n) for n in range(31)] # _G. C. Greubel_, Feb 10 2024

%Y Cf. A129467, A130559.

%K sign,easy

%O 0,4

%A _Wolfdieter Lang_, May 04 2007