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%I #8 Sep 08 2022 08:45:41
%S 1,1,1,1,11,1,1,21,21,1,1,31,66,31,1,1,41,136,136,41,1,1,51,231,362,
%T 231,51,1,1,61,351,755,755,351,61,1,1,71,496,1361,1870,1361,496,71,1,
%U 1,81,666,2226,3906,3906,2226,666,81,1,1,91,861,3396,7266,9282,7266,3396,861,91,1
%N Triangle, read by rows, T(n,k,m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1), with m = 3.
%C Row sums are: {1, 2, 13, 44, 130, 356, 928, 2336, 5728, 13760, 32512, ...}.
%H G. C. Greubel, <a href="/A157171/b157171.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k,m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1), with m = 3.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 11, 1;
%e 1, 21, 21, 1;
%e 1, 31, 66, 31, 1;
%e 1, 41, 136, 136, 41, 1;
%e 1, 51, 231, 362, 231, 51, 1;
%e 1, 61, 351, 755, 755, 351, 61, 1;
%e 1, 71, 496, 1361, 1870, 1361, 496, 71, 1;
%e 1, 81, 666, 2226, 3906, 3906, 2226, 666, 81, 1;
%e 1, 91, 861, 3396, 7266, 9282, 7266, 3396, 861, 91, 1;
%p T:= proc(n, k, m) option remember;
%p if k=0 and n=0 then 1
%p else (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1)
%p fi; end:
%p seq(seq(T(n, k, 3), k=0..n), n=0..10); # _G. C. Greubel_, Nov 29 2019
%t T[n_, k_, m_]:= If[n==0 && k==0, 1, (m*(n-k)+1)*Binomial[n-1, k-1] + (m*k+1)*Binomial[n-1, k] + m*k*(n-k)*Binomial[n-2, k-1]]; Table[T[n, k, 3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Nov 29 2019 *)
%o (PARI) T(n,k,m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1); \\ _G. C. Greubel_, Nov 29 2019
%o (Magma) m:=3; [(m*(n-k)+1)*Binomial(n-1, k-1) + (m*k+1)* Binomial(n-1, k) + m*k*(n-k)*Binomial(n-2, k-1): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 29 2019
%o (Sage) m=3; [[(m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1) for k in (0..n)] for n in [0..10]] # _G. C. Greubel_, Nov 29 2019
%o (GAP) m:=3;; Flat(List([0..10], n-> List([0..n], k-> (m*(n-k)+1)*Binomial(n-1, k-1) + (m*k+1)* Binomial(n-1, k) + m*k*(n-k)*Binomial(n-2, k-1) ))); # _G. C. Greubel_, Nov 29 2019
%Y Cf. A157169 (m=1), A157170 (m=2), this sequence (m=3).
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 24 2009
%E Edited by _G. C. Greubel_, Nov 29 2019
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