A371685 Triangle read by rows: T(n, k) = n! * Sum_{j=0..n-1} binomial(k - 1, j) / (j + 1).
0, 1, 1, 1, 2, 3, 5, 6, 9, 14, 14, 24, 36, 56, 90, 94, 120, 180, 280, 450, 744, 444, 720, 1080, 1680, 2700, 4464, 7560, 3828, 5040, 7560, 11760, 18900, 31248, 52920, 91440, 25584, 40320, 60480, 94080, 151200, 249984, 423360, 731520, 1285200
Offset: 0
Examples
Triangle starts: [0] 0; [1] 1, 1; [2] 1, 2, 3; [3] 5, 6, 9, 14; [4] 14, 24, 36, 56, 90; [5] 94, 120, 180, 280, 450, 744; [6] 444, 720, 1080, 1680, 2700, 4464, 7560; [7] 3828, 5040, 7560, 11760, 18900, 31248, 52920, 91440;
Programs
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Maple
T := (n, k) -> local j; n!*add(binomial(k-1, j)/(j + 1), j = 0..n-1): T := (n, k) -> local j; n!*ifelse(n = 0, 0, ifelse(k=0, add(-(-1)^j/j, j = 1..n), (2^k - 1) / k)): seq(print(seq(T(n, k), k = 0..n)), n = 0..7);
Formula
Restricted to the range 1 <= k <= n: T(n, k) = n!*(2^k - 1)/k.