cp's OEIS Frontend

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

%I A129463 #12 Feb 14 2024 06:50:49
%S A129463 1,0,-1,-4,-39,-680,-18425,-713820,-37390255,-2543067280,
%T A129463 -217799766225,-22928327328500,-2909576503498775,-437960283393276600,
%U A129463 -77145678498655849225,-15720035935018890359500,-3668950667796545284209375,-972327797466833893742228000
%N A129463 Row sums of triangle A129462 (v=2 member of a certain family).
%C A129463 See A129462 for the M. Bruschi et al. reference.
%H A129463 G. C. Greubel, <a href="/A129463/b129463.txt">Table of n, a(n) for n = 0..250</a>
%F A129463 a(n) = Sum_{k=0..n} A129462(n,k), n >= 0.
%F A129463 From _Vaclav Kotesovec_, Aug 24 2016: (Start)
%F A129463 a(n) = 2*(n-2)*(n-1)*a(n-1) - (n-3)^2*(n-1)^2*a(n-2).
%F A129463 a(n) ~ c * n^(2*n+(sqrt(5)-3)/2) / exp(2*n), where c = -2.3203776630375605070105975273368548459...
%F A129463 (End)
%t A129463 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 *)
%t A129463 A129463[n_]:= A129463[n]= Sum[T[n,k], {k,0,n}];
%t A129463 Table[A129463[n], {n,0,40}] (* _G. C. Greubel_, Feb 08 2024 *)
%o A129463 (SageMath)
%o A129463 @CachedFunction
%o A129463 def T(n,k): # T = A129462
%o A129463     if (k<0 or k>n): return 0
%o A129463     elif (n==0): return 1
%o A129463     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)
%o A129463 def A129463(n): return sum(T(n,k) for k in range(n+1))
%o A129463 [A129463(n) for n in range(41)] # _G. C. Greubel_, Feb 08 2024
%Y A129463 Cf. A129458 (row sums v=1 member), A129462.
%K A129463 sign,easy
%O A129463 0,4
%A A129463 _Wolfdieter Lang_, May 04 2007