%I #10 Feb 09 2021 21:40:14
%S 1,1,1,1,-13,1,1,-74,-74,1,1,-278,-588,-278,1,1,-881,-3086,-3086,-881,
%T 1,1,-2539,-13207,-22097,-13207,-2539,1,1,-6884,-49724,-124694,
%U -124694,-49724,-6884,1,1,-17884,-171184,-600424,-900892,-600424,-171184,-17884,1
%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 = 2, 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="/A174731/b174731.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 = 2.
%F From _G. C. Greubel_, Feb 09 2021: (Start)
%F T(n, k, 2) = (1-2^n)*(A001263(n,k) - 1) + 1.
%F Sum_{k=1..n} T(n, k, 2) = 2^n * n + (1 - 2^n)*A000108(n). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -13, 1;
%e 1, -74, -74, 1;
%e 1, -278, -588, -278, 1;
%e 1, -881, -3086, -3086, -881, 1;
%e 1, -2539, -13207, -22097, -13207, -2539, 1;
%e 1, -6884, -49724, -124694, -124694, -49724, -6884, 1;
%e 1, -17884, -171184, -600424, -900892, -600424, -171184, -17884, 1;
%e 1, -45011, -551396, -2576936, -5412692, -5412692, -2576936, -551396, -45011, 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, 2], {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,2) 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,2): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 09 2021
%Y Cf. A000108, A001263.
%Y Cf. A000012 (q=1), this sequence (q=2), A174732 (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