OFFSET
1,2
COMMENTS
Square matrices A(d) from the sequence are related to the inverse of Vandermonde matrices of the type V(1-s,...,d-s)[i,j] = (i-s)^(j-1), for 1 <= i,j <= d .
In particular, if s = 0, A(d) = [V(1,...,d)]^(-1) * (d-1)!.
A(d) can be generated using corresponding square blocks in A335442.
LINKS
ProofWiki, Inverse of Vandermonde Matrix.
Eric Weisstein's World of Mathematics, K-Function.
Wikipedia, Vandermonde matrix.
FORMULA
For d >= 1, if A(d) denotes the d-th square block from the sequence:
.
(1)
If B(d) denotes the corresponding square block from A335442:
A(d)[i,j] = B(d)[d-j+1, d-i+1] * binomial(d-1,j-1) * (-1)^(i+j, for 1 <= i,j <= d
.
(2)
If V(1,...,d) denotes the d-dimensional integer Vandermonde matrix
V(1,...,d)[i,j] = i^(j-1), for 1 <= i,j <= d :
A(d) / (d-1)! = [V(1,...,d)]^(-1) ,
or equivalently, as integer formula:
V(1,...,d) * A(d) = I(d) * (d-1)!
Here, I(d) denotes the d-dimensional identity matrix
.
(3)
More generally, for s = ...,-2,-1,0,1,2,...
If V(1-s,...,d-s) denotes the d-dimensional integer Vandermonde matrix
V(1-s,...,d-s)[i,j] = (i-s)^(j-1), for 1 <= i,j <= d :
T(d,s) * A(d) / (d-1)! = [V(1-s,...,d-s)]^(-1) ,
or equivalently, as integer formula:
V(1-s,...,d-s) * T(d,s) * A(d) = I(d) * (d-1)!
Here, T(d,s) denotes the d-dimensional upper triangular matrix
T(d,s)[i,j] = binomial(j-1,i-1) * s^(j-i) if i <= j
T(d,s)[i,j] = 0 if i > j
.
(4)
determinant[A(d)] = K(d) = A002109(d)
Here, K() denotes the K-function. K(d+1) equals the d-th hyperfactorial.
.
(5)
Row and column sums amount to
Sum_{j=1..d} A(d)[i,j] = delta(i,1) * (d-1)!
Sum_{i=1..d} A(d)[i,j] = delta(j,1) * (d-1)!
Here, delta(i,j) denotes the Kronecker delta.
EXAMPLE
Matrices begin:
d=1: 1,
.
d=2: 2, -1
-1, 1
.
d=3: 6, -6, 2
-5, 8, -3
1, -2, 1
.
d=4: 24, -36, 24, -6
-26, 57, -42, 11
9, -24, 21, -6
-1, 3, -3, 1 .
.
For example, let d = 3:
.
| 6 -6 2 |
A(3) = | -5 8 -3 |
| 1 -2 1 |
.
| 1 1 1 |
V(1,2,3) = | 1 2 4 |
| 1 3 9 |
.
| 2 0 0 |
V(1,2,3) * A(3) = | 0 2 0 |
| 0 0 2 |
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Andreas B. G. Blobel, Sep 13 2021
STATUS
approved