A174731 - OEIS

%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