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A157174 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. 2

%I

%S 1,1,1,1,5,1,1,9,9,1,1,13,18,13,1,1,17,28,28,17,1,1,21,39,38,39,21,1,

%T 1,25,51,35,35,51,25,1,1,29,64,11,-50,11,64,29,1,1,33,78,-42,-294,

%U -294,-42,78,33,1,1,37,93,-132,-798,-1218,-798,-132,93,37,1

%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, 7, 20, 46, 92, 160, 224, 160, -448, -28166, ...}.

%H G. C. Greubel, <a href="/A157174/b157174.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, 5, 1;

%e 1, 9, 9, 1;

%e 1, 13, 18, 13, 1;

%e 1, 17, 28, 28, 17, 1;

%e 1, 21, 39, 38, 39, 21, 1;

%e 1, 25, 51, 35, 35, 51, 25, 1;

%e 1, 29, 64, 11, -50, 11, 64, 29, 1;

%e 1, 33, 78, -42, -294, -294, -42, 78, 33, 1;

%e 1, 37, 93, -132, -798, -1218, -798, -132, 93, 37, 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. A157172 (m=2), this sequence (m=3).

%K sign,tabl

%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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Last modified January 19 03:54 EST 2020. Contains 331031 sequences. (Running on oeis4.)