A172300 Triangle T(n, k, q) = ((1-q)/(1-q^k))*c(n-1, q)*c(n, q)/(c(k-1,q)^2*c(n-k,q)*c(n-k+1, q)), where c(n, q) = Product_{j=1..n} (1-q^j) and q = 3, read by rows.

1, 1, 1, 1, 13, 1, 1, 130, 130, 1, 1, 1210, 12100, 1210, 1, 1, 11011, 1024870, 1024870, 11011, 1, 1, 99463, 84245161, 784128037, 84245161, 99463, 1, 1, 896260, 6857285260, 580812061522, 580812061522, 6857285260, 896260, 1, 1, 8069620, 556344432400, 425659125229240, 3873498039586084, 425659125229240, 556344432400, 8069620, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Formula

Triangle T(n, k, q) = ((1-q)/(1-q^k))*c(n-1, q)*c(n, q)/(c(k-1,q)^2*c(n-k,q)*c(n-k+1, q)), where c(n, q) = Product_{j=1..n} (1-q^j) and q = 3.

Example

Triangle begins as:
  1;
  1,      1;
  1,     13,          1;
  1,    130,        130,            1;
  1,   1210,      12100,         1210,            1;
  1,  11011,    1024870,      1024870,        11011,          1;
  1,  99463,   84245161,    784128037,     84245161,      99463,      1;
  1, 896260, 6857285260, 580812061522, 580812061522, 6857285260, 896260, 1;

Mathematica

c[n_, q_]:= QPochhammer[q, q, n];
T[n_, k_, q_]:= ((1-q)/(1-q^k))*c[n-1, q]*c[n, q]/(c[k-1, q]^2*c[n-k, q]*c[n-k+1, q]);
Table[T[n, k, 3], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, May 07 2021 *)

Prog

(Sage)
from sage.combinat.q_analogues import q_pochhammer
def c(n, q): return q_pochhammer(n, q, q)
def T(n, k, q): return ((1-q)/(1-q^k))*c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q))
[[T(n, k, 3) for k in (1..n)] for n in (1..10)] # G. C. Greubel, May 07 2021

Crossrefs

Cf. A156916 (q=2), this sequence (q=3), A172301 (q=4), A172302 (q=5).

Sequence in context: A353952 A340432 A156539 * A022176 A188646 A174791

Adjacent sequences: A172297 A172298 A172299 * A172301 A172302 A172303

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 31 2010

Extensions

Edited by G. C. Greubel, May 07 2021

Status

approved