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%I #15 Jun 01 2026 13:43:04
%S 1,1,2,1,11,4,1,35,85,8,1,85,725,575,16,1,175,3850,11935,3661,32,1,
%T 322,15190,129360,173901,22631,64,1,546,48804,929334,3665151,2360295,
%U 137845,128,1,870,134820,5012406,46149327,93355395,30673525,833375,256,1,1320,331650,21864150,405247689,2000578965,2216707900,387820895,5019421,512
%N 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)|.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Falling_and_rising_factorials">Falling and rising factorials</a>
%F 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).
%e Array starts:
%e [0] 1, 2, 4, 8, 16, 32, 64, ... [A000079]
%e [1] 1, 11, 85, 575, 3661, 22631, 137845, ... [A001240]
%e [2] 1, 35, 725, 11935, 173901, 2360295, 30673525, ...
%e [3] 1, 85, 3850, 129360, 3665151, 93355395, 2216707900, ...
%e [4] 1, 175, 15190, 929334, 46149327, 2000578965, 79000290160, ...
%e [5] 1, 322, 48804, 5012406, 405247689, 27912905328, 1717078544896, ...
%e [A000012,A000914,...]
%p det := LinearAlgebra:-Determinant: StirlingCycle := abs@Stirling1:
%p m := (n, i, j) -> StirlingCycle(n + i + 2, n + j):
%p M := (n, k) -> Matrix(k, k, (i, j) -> m(n, i, j)): A := (n, k) -> det(M(n, k)):
%p seq(seq(A(n - k, k), k = 0..n), n = 0..9); # _Peter Luschny_, May 31 2026
%o (PARI) A(n, k) = matdet(matrix(k, k, i, j, abs(stirling(n+i+2, n+j, 1))))
%Y Cf. A000012, A000079, A000914, A001240, A132393.
%K nonn,tabl
%O 0,3
%A _Mikhail Kurkov_, May 29 2026