A350200 Array read by antidiagonals: T(n,k) is the determinant of the Hankel matrix of the 2*n-1 consecutive primes starting at the k-th prime, n >= 0, k >= 1.

1, 1, 2, 1, 3, 1, 1, 5, -4, -2, 1, 7, 6, 12, 0, 1, 11, -30, -72, 144, 288, 1, 13, 18, 72, 0, 576, -1728, 1, 17, -42, -72, 288, 1152, -7104, -26240, 1, 19, 30, -96, 144, -1248, -11712, 45248, 222272, 1, 23, 22, -188, 488, -112, -11360, 21184, 450432, 1636864

Offset: 0

Author

Pontus von Brömssen, Dec 19 2021

Keywords

• sign
• tabl

Examples

			Array begins:
  n\k|      1      2       3       4       5       6       7       8
  ---+--------------------------------------------------------------
   0 |      1      1       1       1       1       1       1       1
   1 |      2      3       5       7      11      13      17      19
   2 |      1     -4       6     -30      18     -42      30      22
   3 |     -2     12     -72      72     -72     -96    -188    -480
   4 |      0    144       0     288     144     488    1800    2280
   5 |    288    576    1152   -1248    -112    4432   -1552   15952
   6 |  -1728  -7104  -11712  -11360  -10816   29952  -89152  -57088
   7 | -26240  45248   21184 -103168  -43264 -605440 -379264  271552
   8 | 222272 450432 1068800 2022912 3927552 5399552 6315904 6861312
T(3,2) = 12, the determinant of the Hankel matrix
  [3  5  7]
  [5  7 11]
  [7 11 13].
		

Links

• Wikipedia, Hankel matrix

Crossrefs

Cf. A350201.

Cf. A000012 (row n = 0), A000040 (row n = 1), A056221 (row n = 2 with opposite sign), A024356 (column k = 1), A071543 (column k = 2).

Programs

Python

from sympy import Matrix,prime,nextprime
def A350200(n,k):
    p = [prime(k)] if n > 0 else []
    for i in range(2*n-2): p.append(nextprime(p[-1]))
    return Matrix(n,n,lambda i,j:p[i+j]).det()

Extensions

Offset corrected by Pontus von Brömssen, Aug 25 2022