A347781 Sequence composed of consecutive square matrices A(d) with dimension d=1,2,3,... Matrix elements are arranged by increasing row index i, and, within fixed i, by increasing column index j. Each block A(d) is related to the inverse of a class of integer Vandermonde matrices.

1, 2, -1, -1, 1, 6, -6, 2, -5, 8, -3, 1, -2, 1, 24, -36, 24, -6, -26, 57, -42, 11, 9, -24, 21, -6, -1, 3, -3, 1, 120, -240, 240, -120, 24, -154, 428, -468, 244, -50, 71, -236, 294, -164, 35, -14, 52, -72, 44, -10, 1, -4, 6, -4, 1, 720, -1800, 2400, -1800, 720, -120, -1044, 3510, -5080, 3960, -1620, 274, 580, -2305, 3720, -3070, 1300, -225, -155, 685, -1210, 1070, -475, 85, 20, -95, 180, -170, 80, -15, -1, 5, -10, 10, -5, 1

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

Table of n, a(n) for n=1..91.

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

Cf. A335442, A002109.

Sequence in context: A187783 A089759 A325970 * A088152 A049270 A025269

Adjacent sequences: A347778 A347779 A347780 * A347782 A347783 A347784

Keyword

sign,tabf

Author

Andreas B. G. Blobel, Sep 13 2021

Status

approved