A355401 Triangle read by rows: T(n, k) = Sum_{i=1..n-k} inverse-q-binomial(n-k-1, i-1) * q-binomial(n-2+i, n-2) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0, here q = 2.

1, 0, 1, 0, 1, 1, 0, 4, 3, 1, 0, 64, 28, 7, 1, 0, 4096, 960, 140, 15, 1, 0, 1048576, 126976, 9920, 620, 31, 1, 0, 1073741824, 66060288, 2666496, 89280, 2604, 63, 1, 0, 4398046511104, 136365211648, 2796552192, 48377856, 755904, 10668, 127, 1

Offset: 0

Author

Werner Schulte, Jun 30 2022

Keywords

• nonn
• easy
• tabl

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.

Examples

			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

Cf. A022166, A053763 (column 1), A135950.

Formula

Conjecture: T(n+1, 1) = A053763(n) = 2^(n*n - n) for n >= 0.

Conjecture: T(n, k) = 2^((n-k-1) * (n-k)) * A022166(n-1, k-1) for 0 < k <= n, and T(n, 0) = 0^n for n >= 0.

Conjecture: Define g(n) = -Sum_{i=0..n-1} A022166(n, i) * g(i) * T(n+1-i, 1) for n > 0 with g(0) = 1. Then matrix inverse R = T^(-1) is given by R(n, 0) = 0^n for n >= 0, and R(n, k) = g(n-k) * A022166(n-1, k-1) for 0 < k <= n, and g(n) = R(n+1, 1) for n >= 0.