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