A172301 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 = 4, read by rows.

1, 1, 1, 1, 21, 1, 1, 357, 357, 1, 1, 5797, 98549, 5797, 1, 1, 93093, 25698101, 25698101, 93093, 1, 1, 1490853, 6608951349, 107316781429, 6608951349, 1490853, 1, 1, 23859109, 1693829725237, 441691010116213, 441691010116213, 1693829725237, 23859109, 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 = 4.

Example

Triangle begins as:
  1;
  1,       1;
  1,      21,          1;
  1,     357,        357,            1;
  1,    5797,      98549,         5797,          1;
  1,   93093,   25698101,     25698101,      93093,       1;
  1, 1490853, 6608951349, 107316781429, 6608951349, 1490853, 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, 4], {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, 4) for k in (1..n)] for n in (1..10)] # G. C. Greubel, May 07 2021

Crossrefs

Cf. A156916 (q=2), A172300 (q=3), this sequence (q=4), A172302 (q=5).

Sequence in context: A190581 A350999 A291073 * A022184 A176643 A015147

Adjacent sequences: A172298 A172299 A172300 * A172302 A172303 A172304

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 31 2010

Extensions

Edited by G. C. Greubel, May 07 2021

Status

approved