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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.
3
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
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 31 2010
EXTENSIONS
Edited by G. C. Greubel, May 07 2021
STATUS
approved