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A154231 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1), read by rows.

%I A154231 #16 Mar 06 2021 03:35:31
%S A154231 1,1,1,1,278,1,1,1579,1579,1,1,6005,1233308,6005,1,1,18207,20504692,
%T A154231 20504692,18207,1,1,47216,194715939,35816807848,194715939,47216,1,1,
%U A154231 108993,1319518787,1302709376779,1302709376779,1319518787,108993,1,1,229819,7024500980,24830582225241,4330171226988158,24830582225241,7024500980,229819,1
%N A154231 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1), read by rows.
%C A154231 Row sums are: {1, 2, 280, 3160, 1245320, 41045800, 36206334160, ...}.
%C A154231 The row sums of this class of sequences (see cross-references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = (n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12 = A000539(n+1). - _G. C. Greubel_, Mar 02 2021
%H A154231 G. C. Greubel, <a href="/A154231/b154231.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A154231 T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.
%e A154231 Triangle begins as:
%e A154231   1;
%e A154231   1,      1;
%e A154231   1,    278,          1;
%e A154231   1,   1579,       1579,             1;
%e A154231   1,   6005,    1233308,          6005,             1;
%e A154231   1,  18207,   20504692,      20504692,         18207,          1;
%e A154231   1,  47216,  194715939,   35816807848,     194715939,      47216,      1;
%e A154231   1, 108993, 1319518787, 1302709376779, 1302709376779, 1319518787, 108993, 1;
%p A154231 T:= proc(n, k) option remember;
%p A154231       if k=0 or k=n then 1
%p A154231     else T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2+6*n+3)/12)*T(n-2, k-1)
%p A154231       fi; end:
%p A154231 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 02 2021
%t A154231 T[n_, k_]:= T[n,k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + ((n+1)^2*(n+2)^2*(2*n^2+6*n+3)/12)*T[n-2, k-1] ];
%t A154231 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 2021 *)
%o A154231 (Sage)
%o A154231 def f(n): return binomial(n+2,2)^2*(2*n^2+6*n+3)/3
%o A154231 def T(n,k):
%o A154231     if (k==0 or k==n): return 1
%o A154231     else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
%o A154231 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 02 2021
%o A154231 (Magma)
%o A154231 f:= func< n | Binomial(n+2,2)^2*(2*n^2+6*n+3)/3 >;
%o A154231 function T(n,k)
%o A154231   if k eq 0 or k eq n then return 1;
%o A154231   else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
%o A154231   end if; return T;
%o A154231 end function;
%o A154231 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 02 2021
%Y A154231 Cf. A154227, A154228, A154229, A154230, A154233.
%Y A154231 Cf. A000539 (powers of 5).
%K A154231 nonn,tabl,easy
%O A154231 0,5
%A A154231 _Roger L. Bagula_, Jan 05 2009
%E A154231 Edited by _G. C. Greubel_, Mar 02 2021