A335442 -id:A335442 - cp's OEIS frontend

A126682 Square pyramid giving coefficients of Carlo Wood's polynomials, read by successive slices, each slice being read row by row.

1, 1, 1, 1, 3, 1, 3, 2, 2, 9, 7, 1, 6, 11, 1, 6, 11, 6, 3, 22, 45, 26, 3, 26, 69, 46, 1, 10, 35, 50, 1, 10, 35, 50, 24, 4, 45, 170, 255, 126, 6, 75, 320, 525, 274, 4, 55, 270, 545, 326, 1, 15, 85, 225, 274, 1, 15, 85, 225, 274, 120, 5, 81, 485, 1335, 1670, 744, 10

Offset: 1

N. J. A. Sloane, Feb 14 2007

There is no standard method for converting a pyramid of numbers to a sequence. This seems as good a solution as any.
See the link for further information and more terms.
The first row of each slice seems to coincide with the first row of each slice of A335442. That row from the n-th slice seems to be the coefficients of the polynomial (x+1) * ... * (x+n-1), i.e., the reversed row n-1 of A130534. - Andrey Zabolotskiy, Jun 26 2022

			Slice 1:
  1
Slice 2:
  1 1
  1 3
Slice 3:
  1  3  2
  2  9  7
  1  6 11
Slice 4:
  1  6 11  6
  3 22 45 26
  3 26 69 46
  1 10 35 50
Note that in Part 4 of the linked file, the order of the rows is reversed, while in its Part 1 the order of both rows and columns is reversed.

N. J. A. Sloane, Notes on Carlo Wood's Polynomials

Cf. A126671, A130534, A335442.

The sole term 1 of slice 1 inserted by Andrey Zabolotskiy, Jun 26 2022

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.

Original entry on oeis.org

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

Andreas B. G. Blobel, Sep 13 2021

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.

			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 |

ProofWiki, Inverse of Vandermonde Matrix.
Eric Weisstein's World of Mathematics, K-Function.
Wikipedia, Vandermonde matrix.

Cf. A335442, A002109.

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.

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A126682 Square pyramid giving coefficients of Carlo Wood's polynomials, read by successive slices, each slice being read row by row.

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