A129463 Row sums of triangle A129462 (v=2 member of a certain family).

1, 0, -1, -4, -39, -680, -18425, -713820, -37390255, -2543067280, -217799766225, -22928327328500, -2909576503498775, -437960283393276600, -77145678498655849225, -15720035935018890359500, -3668950667796545284209375, -972327797466833893742228000

Offset: 0

Author

Wolfdieter Lang, May 04 2007

Keywords

• sign
• easy

Comments

See A129462 for the M. Bruschi et al. reference.

Links

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

Crossrefs

Cf. A129458 (row sums v=1 member), A129462.

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0, 1, (2*(n-1)*(n-2) - 1)*T[n-1,k] -((n-1)*(n-3))^2*T[n-2,k] +T[n-1,k-1] ]]; (* T=A129462 *)
A129463[n_]:= A129463[n]= Sum[T[n,k], {k,0,n}];
Table[A129463[n], {n,0,40}] (* G. C. Greubel, Feb 08 2024 *)

SageMath

@CachedFunction
def T(n,k): # T = A129462
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return (2*(n-1)*(n-2)-1)*T(n-1,k) - ((n-1)*(n-3))^2*T(n-2,k) + T(n-1,k-1)
def A129463(n): return sum(T(n,k) for k in range(n+1))
[A129463(n) for n in range(41)] # G. C. Greubel, Feb 08 2024

Formula

a(n) = Sum_{k=0..n} A129462(n,k), n >= 0.

From Vaclav Kotesovec, Aug 24 2016: (Start)

a(n) = 2*(n-2)*(n-1)*a(n-1) - (n-3)^2*(n-1)^2*a(n-2).

a(n) ~ c * n^(2*n+(sqrt(5)-3)/2) / exp(2*n), where c = -2.3203776630375605070105975273368548459...

(End)