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%I #9 Feb 09 2021 21:40:22
%S 1,1,1,1,-51,1,1,-399,-399,1,1,-2177,-4597,-2177,1,1,-10191,-35671,
%T -35671,-10191,1,1,-43719,-227343,-380363,-227343,-43719,1,1,-177119,
%U -1279199,-3207839,-3207839,-1279199,-177119,1,1,-688869,-6593469,-23126349,-34699365,-23126349,-6593469,-688869,1
%N Triangle T(n, k, q) = (1-q^n)*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - (1-q^n) + 1, for q = 3, read by rows.
%C From _G. C. Greubel_, Feb 09 2021: (Start)
%C The triangle coefficients are connected to the Narayana numbers by T(n, k, q) = (1-q^n)*(A001263(n, k) - 1) + 1, for varying q values.
%C The row sums of this class of sequences, for varying q, is given by Sum_{k=1..n} T(n, k, q) = q^n * n + (1 - q^n)*C_{n}, where C_{n} are the Catalan numbers (A000108). (End)
%H G. C. Greubel, <a href="/A174732/b174732.txt">Rows n = 1..100 of the triangle, flattened</a>
%F T(n, k, q) = (1-q^n)*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - (1-q^n) + 1, for q = 3.
%F From _G. C. Greubel_, Feb 09 2021: (Start)
%F T(n, k, 3) = (1-3^n)*(A001263(n,k) - 1) + 1.
%F Sum_{k=1..n} T(n, k, 3) = 3^n * n + (1 - 3^n)*A000108(n). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -51, 1;
%e 1, -399, -399, 1;
%e 1, -2177, -4597, -2177, 1;
%e 1, -10191, -35671, -35671, -10191, 1;
%e 1, -43719, -227343, -380363, -227343, -43719, 1;
%e 1, -177119, -1279199, -3207839, -3207839, -1279199, -177119, 1;
%e 1, -688869, -6593469, -23126349, -34699365, -23126349, -6593469, -688869, 1;
%t T[n_, k_, q_]:= 1 + (1-q^n)*(1/k)*(Binomial[n-1, k-1]*Binomial[n, k-1] - k);
%t Table[T[n, k, 3], {n, 12}, {k, n}]//Flatten
%o (Sage)
%o def T(n,k,q): return 1 + (1-q^n)*(1/k)*(binomial(n-1, k-1)*binomial(n, k-1) - k)
%o flatten([[T(n,k,3) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 09 2021
%o (Magma)
%o T:= func< n,k,q | 1 +(1-q^n)*(1/k)*(Binomial(n-1, k-1)*Binomial(n, k-1) - k) >;
%o [T(n,k,3): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 09 2021
%Y Cf. A000108, A001263.
%Y Cf. A000012 (q=1), A174731 (q=2), this sequence (q=3), A174733 (q=4).
%K sign,tabl
%O 1,5
%A _Roger L. Bagula_, Mar 28 2010
%E Edited by _G. C. Greubel_, Feb 09 2021
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