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A154229 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1), read by rows.

%I A154229 #12 Mar 04 2021 02:36:21
%S A154229 1,1,1,1,38,1,1,139,139,1,1,365,8828,365,1,1,807,70492,70492,807,1,1,
%T A154229 1592,357459,7062136,357459,1592,1,1,2889,1404923,98777227,98777227,
%U A154229 1404923,2889,1,1,4915,4631612,824036625,14498379854,824036625,4631612,4915,1
%N A154229 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1), read by rows.
%C A154229 Row sums are: {1, 2, 40, 280, 9560, 142600, 7780240, 200370080, 16155726160, ...}.
%C A154229 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) = binomial(n+2, 2)^2 = A000537(n+1). - _G. C. Greubel_, Mar 02 2021
%H A154229 G. C. Greubel, <a href="/A154229/b154229.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A154229 T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.
%e A154229 Triangle begins as:
%e A154229   1;
%e A154229   1,    1;
%e A154229   1,   38,       1;
%e A154229   1,  139,     139,         1;
%e A154229   1,  365,    8828,       365,           1;
%e A154229   1,  807,   70492,     70492,         807,         1;
%e A154229   1, 1592,  357459,   7062136,      357459,      1592,       1;
%e A154229   1, 2889, 1404923,  98777227,    98777227,   1404923,    2889,    1;
%e A154229   1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 1;
%p A154229 T:= proc(n, k) option remember;
%p A154229       if k=0 or k=n then 1
%p A154229     else T(n-1, k) + T(n-1, k-1) + binomial(n+2,2)^2*T(n-2, k-1)
%p A154229       fi; end:
%p A154229 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 02 2021
%t A154229 T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + Binomial[n+2, 2]^2*T[n-2, k-1]];
%t A154229 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 2021 *)
%o A154229 (Sage)
%o A154229 def f(n): return binomial(n+2,2)^2
%o A154229 def T(n,k):
%o A154229     if (k==0 or k==n): return 1
%o A154229     else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
%o A154229 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 02 2021
%o A154229 (Magma)
%o A154229 f:= func< n | Binomial(n+2,2)^2 >;
%o A154229 function T(n,k)
%o A154229   if k eq 0 or k eq n then return 1;
%o A154229   else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
%o A154229   end if; return T;
%o A154229 end function;
%o A154229 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 02 2021
%Y A154229 Cf. A154227, A154228, A154230, A154231, A154233.
%Y A154229 Cf. A000537.
%K A154229 nonn,tabl
%O A154229 0,5
%A A154229 _Roger L. Bagula_, Jan 05 2009
%E A154229 Edited by _G. C. Greubel_, Mar 02 2021