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%I #5 Mar 02 2021 16:18:12
%S 1,1,1,1,8,1,1,19,19,1,1,35,158,35,1,1,57,592,592,57,1,1,86,1629,5608,
%T 1629,86,1,1,123,3767,28549,28549,3767,123,1,1,169,7760,105621,309458,
%U 105621,7760,169,1,1,225,14694,320566,1985274,1985274,320566,14694,225,1
%N Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.
%C Row sums are: {1, 2, 10, 40, 230, 1300, 9040, 64880, 536560, 4641520, ...}.
%C The row sums of this class of sequences (see Cf section) 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). - _G. C. Greubel_, Mar 02 2021
%H G. C. Greubel, <a href="/A154227/b154227.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)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 8, 1;
%e 1, 19, 19, 1;
%e 1, 35, 158, 35, 1;
%e 1, 57, 592, 592, 57, 1;
%e 1, 86, 1629, 5608, 1629, 86, 1;
%e 1, 123, 3767, 28549, 28549, 3767, 123, 1;
%e 1, 169, 7760, 105621, 309458, 105621, 7760, 169, 1;
%e 1, 225, 14694, 320566, 1985274, 1985274, 320566, 14694, 225, 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)*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]*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)
%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) >;
%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. A154228, A154229, A154230, A154231, A154233.
%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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