A396563 Array read by ascending antidiagonals: A(n, k) is the determinant of the k X k matrix, where the (i, j)-th entry is |Stirling1(n + i + 2, n + j)|.

1, 1, 2, 1, 11, 4, 1, 35, 85, 8, 1, 85, 725, 575, 16, 1, 175, 3850, 11935, 3661, 32, 1, 322, 15190, 129360, 173901, 22631, 64, 1, 546, 48804, 929334, 3665151, 2360295, 137845, 128, 1, 870, 134820, 5012406, 46149327, 93355395, 30673525, 833375, 256, 1, 1320, 331650, 21864150, 405247689, 2000578965, 2216707900, 387820895, 5019421, 512

Offset

0,3

Links

Table of n, a(n) for n=0..54.

Wikipedia, Falling and rising factorials

Formula

Conjecture: B(n, k, m) = (Sum_{j_1=0..n} Sum_{j_2=0..j_1} ... Sum_{j_m=0..j_{m-1}} (-1)^(j_1 + j_2 + ... + j_m) * (Product_{1 <= q < p <= m} (j_q - j_p + p - q)^2) * Product_{t=1..m} (n+t-1)_{j_t} * (n-j_t+t)^(n+k) / (j_t+m-t)!) / (Product_{i=0..m-1} (n+i)!) where B(n, k, m) is the determinant of the k X k matrix with (i, j)-th entry |Stirling1(n+m+i, n+j)|, where (x)_{n} is the falling factorial defined as (x)_{n} = Product_{k=0..n-1} (x-k).

Example

Array starts:
  [0] 1,   2,     4,       8,        16,          32,            64, ... [A000079]
  [1] 1,  11,    85,     575,      3661,       22631,        137845, ... [A001240]
  [2] 1,  35,   725,   11935,    173901,     2360295,      30673525, ...
  [3] 1,  85,  3850,  129360,   3665151,    93355395,    2216707900, ...
  [4] 1, 175, 15190,  929334,  46149327,  2000578965,   79000290160, ...
  [5] 1, 322, 48804, 5012406, 405247689, 27912905328, 1717078544896, ...
  [A000012,A000914,...]

Maple

det := LinearAlgebra:-Determinant: StirlingCycle := abs@Stirling1:
m := (n, i, j) -> StirlingCycle(n + i + 2, n + j):
M := (n, k) -> Matrix(k, k, (i, j) -> m(n, i, j)): A := (n, k) -> det(M(n, k)):
seq(seq(A(n - k, k), k = 0..n), n = 0..9);  # Peter Luschny, May 31 2026

Prog

(PARI) A(n, k) = matdet(matrix(k, k, i, j, abs(stirling(n+i+2, n+j, 1))))

Crossrefs

Cf. A000012, A000079, A000914, A001240, A132393.

Sequence in context: A252158 A285996 A191618 * A306547 A143888 A016546

Adjacent sequences: A396560 A396561 A396562 * A396564 A396566 A396567

Keyword

nonn,tabl

Author

Mikhail Kurkov, May 29 2026

Status

approved