A129458 Row sums of triangle A129065 (v=1 member of a family).

1, 1, 3, 23, 329, 7545, 253195, 11692735, 710944785, 55043460305, 5286546264275, 616743770648775, 85901526469924825, 14079397690024018825, 2682416268746051840475, 587823624532842773747375, 146813897212611204795118625, 41456888496977804292047054625

Offset: 0

Author

Wolfdieter Lang, May 04 2007

Keywords

• nonn
• easy

Comments

See A129065 for the M. Bruschi et al. reference.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A129065.

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0, 1, 2*(n-1)^2*T[n-1,k] - 4*Binomial[n-1,2]^2*T[n-2,k] +T[n-1,k-1] ]]; (* T = A129065 *)
A129458[n_]:= A129458[n]= Sum[T[n,k], {k,0,n}];
Table[A129458[n], {n,0,40}] (* G. C. Greubel, Feb 08 2024 *)

SageMath

@CachedFunction
def T(n,k): # T = A129065
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return 2*(n-1)^2*T(n-1,k) - 4*binomial(n-1,2)^2*T(n-2,k) + T(n-1,k-1)
def A129458(n): return sum(T(n,k) for k in range(n+1))
[A129458(n) for n in range(41)] # G. C. Greubel, Feb 08 2024

Formula

a(n) = Sum_{m=0..n} A129065(n,m).

From Vaclav Kotesovec, Aug 24 2016: (Start)

a(n) = (2*n^2 - 4*n + 3)*a(n-1) - (n-2)^2*(n-1)^2*a(n-2).

a(n) ~ c * n^(2*n+(sqrt(5)-1)/2) / exp(2*n), where c = 6.07482758856838398336112197806575192722726...

(End)