A174732 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.

1, 1, 1, 1, -51, 1, 1, -399, -399, 1, 1, -2177, -4597, -2177, 1, 1, -10191, -35671, -35671, -10191, 1, 1, -43719, -227343, -380363, -227343, -43719, 1, 1, -177119, -1279199, -3207839, -3207839, -1279199, -177119, 1, 1, -688869, -6593469, -23126349, -34699365, -23126349, -6593469, -688869, 1

Offset

1,5

Comments

From G. C. Greubel, Feb 09 2021: (Start)

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.

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)

Links

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Formula

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.

From G. C. Greubel, Feb 09 2021: (Start)

T(n, k, 3) = (1-3^n)*(A001263(n,k) - 1) + 1.

Sum_{k=1..n} T(n, k, 3) = 3^n * n + (1 - 3^n)*A000108(n). (End)

Example

Triangle begins as:
  1;
  1,       1;
  1,     -51,        1;
  1,    -399,     -399,         1;
  1,   -2177,    -4597,     -2177,         1;
  1,  -10191,   -35671,    -35671,    -10191,         1;
  1,  -43719,  -227343,   -380363,   -227343,    -43719,        1;
  1, -177119, -1279199,  -3207839,  -3207839,  -1279199,  -177119,       1;
  1, -688869, -6593469, -23126349, -34699365, -23126349, -6593469, -688869, 1;

Mathematica

T[n_, k_, q_]:= 1 + (1-q^n)*(1/k)*(Binomial[n-1, k-1]*Binomial[n, k-1] - k);
Table[T[n, k, 3], {n, 12}, {k, n}]//Flatten

Prog

(Sage)
def T(n, k, q): return 1 + (1-q^n)*(1/k)*(binomial(n-1, k-1)*binomial(n, k-1) - k)
flatten([[T(n, k, 3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021
(Magma)
T:= func< n, k, q | 1 +(1-q^n)*(1/k)*(Binomial(n-1, k-1)*Binomial(n, k-1) - k) >;
[T(n, k, 3): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021

Crossrefs

Cf. A000108, A001263.

Cf. A000012 (q=1), A174731 (q=2), this sequence (q=3), A174733 (q=4).

Sequence in context: A015038 A152515 A111402 * A087408 A255852 A160474

Adjacent sequences: A174729 A174730 A174731 * A174733 A174734 A174735

Keyword

sign,tabl

Author

Roger L. Bagula, Mar 28 2010

Extensions

Edited by G. C. Greubel, Feb 09 2021

Status

approved