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%I #12 Sep 08 2022 08:45:41
%S 1,1,1,1,4,1,1,7,7,1,1,10,14,10,1,1,13,22,22,13,1,1,16,31,32,31,16,1,
%T 1,19,41,35,35,41,19,1,1,22,52,26,-10,26,52,22,1,1,25,64,0,-154,-154,
%U 0,64,25,1,1,28,77,-48,-462,-728,-462,-48,77,28,1
%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 = 2.
%C Row sums are: {1, 2, 6, 16, 36, 72, 128, 192, 192, -128, -1536, ...}.
%H G. C. Greubel, <a href="/A157172/b157172.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 = 2.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 7, 7, 1;
%e 1, 10, 14, 10, 1;
%e 1, 13, 22, 22, 13, 1;
%e 1, 16, 31, 32, 31, 16, 1;
%e 1, 19, 41, 35, 35, 41, 19, 1;
%e 1, 22, 52, 26, -10, 26, 52, 22, 1;
%e 1, 25, 64, 0, -154, -154, 0, 64, 25, 1;
%e 1, 28, 77, -48, -462, -728, -462, -48, 77, 28, 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, 2), 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, 2], {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:=2; [(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=2; [[(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:=2;; 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. this sequence (m=2), A157174 (m=3).
%K tabl,sign
%O 0,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Feb 24 2009
%E Edited by _G. C. Greubel_, Nov 29 2019
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