A172302 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 = 5, read by rows.

1, 1, 1, 1, 31, 1, 1, 806, 806, 1, 1, 20306, 527956, 20306, 1, 1, 508431, 333038706, 333038706, 508431, 1, 1, 12714681, 208533483081, 5253698396331, 208533483081, 12714681, 1, 1, 317886556, 130381488829956, 82245646088465706, 82245646088465706, 130381488829956, 317886556, 1

Offset

1,5

Links

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

Formula

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 = 5.

Example

Triangle begins as:
  1;
  1,        1;
  1,       31,            1;
  1,      806,          806,             1;
  1,    20306,       527956,         20306,            1;
  1,   508431,    333038706,     333038706,       508431,        1;
  1, 12714681, 208533483081, 5253698396331, 208533483081, 12714681, 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, 5], {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, 5) for k in (1..n)] for n in (1..10)] # G. C. Greubel, May 07 2021

Crossrefs

Cf. A156916 (q=2), A172300 (q=3), A172301 (q=4), this sequence (q=5).

Sequence in context: A300656 A239633 A174692 * A103474 A047687 A040953

Adjacent sequences: A172299 A172300 A172301 * A172303 A172304 A172305

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 31 2010

Extensions

Edited by G. C. Greubel, May 07 2021

Status

approved