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