OFFSET
0,8
COMMENTS
The Gaussian or q-binomial coefficients [n, k]_q for 0 <= k <= n are the basis for lower triangular matrices T_q, which are created by an unusual formula. This triangle is the result for q = 2. The general construction is as follows:
For some fixed integer q define the infinite lower triangular matrix M_q by M(q; n, 0) = 0 for n > 0, and M(q; n, n) = 1 for n >= 0, and M(q; n, k) = M(q; n-1, k-1) + q^(k-1) * M(q; n-1, k) for 0 < k < n. Then the matrix inverse I_q = M_q^(-1) exists, and M(q; n, k) = [n-1, k-1]_q for 0 < k <= n. Next define the triangle T_q by T(q; n, k) = Sum_{i=0..n-k} I(q; n-k, i) * M(q; n-1+i, n-1) for 0 < k <= n and T(q; n, 0) = 0^n for n >= 0. For q = 1 see A097805 and for q = 2 see this triangle.
Conjecture: T(q; n+1, 1) = q^(n*n-n) for n >= 0.
Conjecture: T(q; n, k) = q^((n-k-1)*(n-k)) * M(q; n, k) for 0 <= k <= n.
Conjecture: Define g(q; n) = -Sum_{i=0..n-1} [n, i]_q * g(q; i) * T(q; n+1-i, 1) for n > 0 with g(q; 0) = 1. Then the matrix inverse R_q = T_q^(-1) is given by R(q; n, k) = g(q; n-k) * M(q; n, k) for 0 <= k <= n, and g(q; n) = R(q; n+1, 1) for n >= 0.
FORMULA
EXAMPLE
Triangle T(n, k) for 0 <= k <= n starts:
n\k : 0 1 2 3 4 5 6 7 8
==================================================================================
0 : 1
1 : 0 1
2 : 0 1 1
3 : 0 4 3 1
4 : 0 64 28 7 1
5 : 0 4096 960 140 15 1
6 : 0 1048576 126976 9920 620 31 1
7 : 0 1073741824 66060288 2666496 89280 2604 63 1
8 : 0 4398046511104 136365211648 2796552192 48377856 755904 10668 127 1
etc.
Matrix inverse R(n, k) for 0 <= k <= n starts:
n\k : 0 1 2 3 4 5 6 7
===============================================================
0 : 1
1 : 0 1
2 : 0 -1 1
3 : 0 -1 -3 1
4 : 0 -29 -7 -7 1
5 : 0 -2561 -435 -35 -15 1
6 : 0 -814309 -79391 -4495 -155 -31 1
7 : 0 -944455609 -51301467 -1667211 -40455 -651 -63 1
etc.
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Jun 30 2022
STATUS
approved