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A129461 Fourth column (m=3) of triangle A129065.

%I A129461 #11 Feb 08 2024 08:46:04
%S A129461 1,28,908,37896,2036592,138517632,11692594944,1202885199360,
%T A129461 148407122764800,21652192199577600,3690199478509977600,
%U A129461 726862474705593139200,163918208008013340672000
%N A129461 Fourth column (m=3) of triangle A129065.
%C A129461 See A129065 for the M. Bruschi et al. reference.
%H A129461 G. C. Greubel, <a href="/A129461/b129461.txt">Table of n, a(n) for n = 0..245</a>
%F A129461 a(n) = A129065(n+3, 3), n >= 0.
%t A129461 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 A129461 A129461[n_]:= T[n+3,3];
%t A129461 Table[A129461[n], {n,0,40}] (* _G. C. Greubel_, Feb 08 2024 *)
%o A129461 (Magma)
%o A129461 function T(n,k) // T = A129065
%o A129461   if k lt 0 or k gt n then return 0;
%o A129461   elif n eq 0 then return 1;
%o A129461   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 A129461   end if;
%o A129461 end function;
%o A129461 A129461:= func< n | T(n+3, 3) >;
%o A129461 [A129461(n): n in [0..20]]; // _G. C. Greubel_, Feb 08 2024
%o A129461 (SageMath)
%o A129461 @CachedFunction
%o A129461 def T(n,k): # T = A129065
%o A129461     if (k<0 or k>n): return 0
%o A129461     elif (n==0): return 1
%o A129461     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 A129461 def A129461(n): return T(n+3,3)
%o A129461 [A129461(n) for n in range(41)] # _G. C. Greubel_, Feb 08 2024
%Y A129461 Cf. A129065, A129460 (m=2).
%K A129461 nonn,easy
%O A129461 0,2
%A A129461 _Wolfdieter Lang_, May 04 2007