A350200 - OEIS

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.

%I #9 Aug 25 2022 16:22:45

%S 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,

%T 576,-1728,1,17,-42,-72,288,1152,-7104,-26240,1,19,30,-96,144,-1248,

%U -11712,45248,222272,1,23,22,-188,488,-112,-11360,21184,450432,1636864

%N 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>

%e Array begins:

%e n\k| 1 2 3 4 5 6 7 8

%e ---+--------------------------------------------------------------

%e 0 | 1 1 1 1 1 1 1 1

%e 1 | 2 3 5 7 11 13 17 19

%e 2 | 1 -4 6 -30 18 -42 30 22

%e 3 | -2 12 -72 72 -72 -96 -188 -480

%e 4 | 0 144 0 288 144 488 1800 2280

%e 5 | 288 576 1152 -1248 -112 4432 -1552 15952

%e 6 | -1728 -7104 -11712 -11360 -10816 29952 -89152 -57088

%e 7 | -26240 45248 21184 -103168 -43264 -605440 -379264 271552

%e 8 | 222272 450432 1068800 2022912 3927552 5399552 6315904 6861312

%e T(3,2) = 12, the determinant of the Hankel matrix

%e [3 5 7]

%e [5 7 11]

%e [7 11 13].

%o (Python)

%o from sympy import Matrix,prime,nextprime

%o def A350200(n,k):

%o p = [prime(k)] if n > 0 else []

%o for i in range(2*n-2): p.append(nextprime(p[-1]))

%o return Matrix(n,n,lambda i,j:p[i+j]).det()

%Y Cf. A350201.

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

%K sign,tabl

%O 0,3

%A _Pontus von Brömssen_, Dec 19 2021

%E Offset corrected by _Pontus von Brömssen_, Aug 25 2022