A172300 - OEIS

%I #6 May 07 2021 09:17:43

%S 1,1,1,1,13,1,1,130,130,1,1,1210,12100,1210,1,1,11011,1024870,1024870,

%T 11011,1,1,99463,84245161,784128037,84245161,99463,1,1,896260,

%U 6857285260,580812061522,580812061522,6857285260,896260,1,1,8069620,556344432400,425659125229240,3873498039586084,425659125229240,556344432400,8069620,1

%N 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.

%H G. C. Greubel, <a href="/A172300/b172300.txt">Rows n = 1..30 of the triangle, flattened</a>

%F 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.

%e Triangle begins as:

%e   1;

%e   1,      1;

%e   1,     13,          1;

%e   1,    130,        130,            1;

%e   1,   1210,      12100,         1210,            1;

%e   1,  11011,    1024870,      1024870,        11011,          1;

%e   1,  99463,   84245161,    784128037,     84245161,      99463,      1;

%e   1, 896260, 6857285260, 580812061522, 580812061522, 6857285260, 896260, 1;

%t c[n_, q_]:= QPochhammer[q, q, n];

%t 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]);

%t Table[T[n, k, 3], {n,10}, {k,n}]//Flatten (* modified by _G. C. Greubel_, May 07 2021 *)

%o (Sage)

%o from sage.combinat.q_analogues import q_pochhammer

%o def c(n,q): return q_pochhammer(n,q,q)

%o 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))

%o [[T(n,k,3) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, May 07 2021

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

%K nonn,tabl

%O 1,5

%A _Roger L. Bagula_, Jan 31 2010

%E Edited by _G. C. Greubel_, May 07 2021