A174731 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.

1, 1, 1, 1, -13, 1, 1, -74, -74, 1, 1, -278, -588, -278, 1, 1, -881, -3086, -3086, -881, 1, 1, -2539, -13207, -22097, -13207, -2539, 1, 1, -6884, -49724, -124694, -124694, -49724, -6884, 1, 1, -17884, -171184, -600424, -900892, -600424, -171184, -17884, 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 = 2.

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

T(n, k, 2) = (1-2^n)*(A001263(n,k) - 1) + 1.

Sum_{k=1..n} T(n, k, 2) = 2^n * n + (1 - 2^n)*A000108(n). (End)

Example

Triangle begins as:
  1;
  1,      1;
  1,    -13,       1;
  1,    -74,     -74,        1;
  1,   -278,    -588,     -278,        1;
  1,   -881,   -3086,    -3086,     -881,        1;
  1,  -2539,  -13207,   -22097,   -13207,    -2539,        1;
  1,  -6884,  -49724,  -124694,  -124694,   -49724,    -6884,       1;
  1, -17884, -171184,  -600424,  -900892,  -600424,  -171184,  -17884,      1;
  1, -45011, -551396, -2576936, -5412692, -5412692, -2576936, -551396, -45011, 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, 2], {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, 2) 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, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021

Crossrefs

Cf. A000108, A001263.

Cf. A000012 (q=1), this sequence (q=2), A174732 (q=3), A174733 (q=4).

Sequence in context: A108477 A176204 A176492 * A342891 A174694 A353952

Adjacent sequences: A174728 A174729 A174730 * A174732 A174733 A174734

Keyword

sign,tabl

Author

Roger L. Bagula, Mar 28 2010

Extensions

Edited by G. C. Greubel, Feb 09 2021

Status

approved