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

%I A129458 #13 Feb 08 2024 01:45:25
%S A129458 1,1,3,23,329,7545,253195,11692735,710944785,55043460305,
%T A129458 5286546264275,616743770648775,85901526469924825,14079397690024018825,
%U A129458 2682416268746051840475,587823624532842773747375,146813897212611204795118625,41456888496977804292047054625
%N A129458 Row sums of triangle A129065 (v=1 member of a family).
%C A129458 See A129065 for the M. Bruschi et al. reference.
%H A129458 Vaclav Kotesovec, <a href="/A129458/b129458.txt">Table of n, a(n) for n = 0..250</a>
%F A129458 a(n) = Sum_{m=0..n} A129065(n,m).
%F A129458 From _Vaclav Kotesovec_, Aug 24 2016: (Start)
%F A129458 a(n) = (2*n^2 - 4*n + 3)*a(n-1) - (n-2)^2*(n-1)^2*a(n-2).
%F A129458 a(n) ~ c * n^(2*n+(sqrt(5)-1)/2) / exp(2*n), where c = 6.07482758856838398336112197806575192722726...
%F A129458 (End)
%t A129458 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 *)
%t A129458 A129458[n_]:= A129458[n]= Sum[T[n,k], {k,0,n}];
%t A129458 Table[A129458[n], {n,0,40}] (* _G. C. Greubel_, Feb 08 2024 *)
%o A129458 (SageMath)
%o A129458 @CachedFunction
%o A129458 def T(n,k): # T = A129065
%o A129458     if (k<0 or k>n): return 0
%o A129458     elif (n==0): return 1
%o A129458     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)
%o A129458 def A129458(n): return sum(T(n,k) for k in range(n+1))
%o A129458 [A129458(n) for n in range(41)] # _G. C. Greubel_, Feb 08 2024
%Y A129458 Cf. A129065.
%K A129458 nonn,easy
%O A129458 0,3
%A A129458 _Wolfdieter Lang_, May 04 2007