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A371685 Triangle read by rows: T(n, k) = n! * Sum_{j=0..n-1} binomial(k - 1, j) / (j + 1).

%I A371685 #7 Apr 09 2024 10:44:37
%S A371685 0,1,1,1,2,3,5,6,9,14,14,24,36,56,90,94,120,180,280,450,744,444,720,
%T A371685 1080,1680,2700,4464,7560,3828,5040,7560,11760,18900,31248,52920,
%U A371685 91440,25584,40320,60480,94080,151200,249984,423360,731520,1285200
%N A371685 Triangle read by rows: T(n, k) = n! * Sum_{j=0..n-1} binomial(k - 1, j) / (j + 1).
%F A371685 Restricted to the range 1 <= k <= n: T(n, k) = n!*(2^k - 1)/k.
%e A371685 Triangle starts:
%e A371685 [0]    0;
%e A371685 [1]    1,    1;
%e A371685 [2]    1,    2,    3;
%e A371685 [3]    5,    6,    9,    14;
%e A371685 [4]   14,   24,   36,    56,    90;
%e A371685 [5]   94,  120,  180,   280,   450,   744;
%e A371685 [6]  444,  720, 1080,  1680,  2700,  4464,  7560;
%e A371685 [7] 3828, 5040, 7560, 11760, 18900, 31248, 52920, 91440;
%p A371685 T := (n, k) -> local j; n!*add(binomial(k-1, j)/(j + 1), j = 0..n-1):
%p A371685 T := (n, k) -> local j; n!*ifelse(n = 0, 0, ifelse(k=0, add(-(-1)^j/j, j = 1..n), (2^k - 1) / k)):
%p A371685 seq(print(seq(T(n, k), k = 0..n)), n = 0..7);
%Y A371685 Cf. A029767 (main diagonal), A024167 (column 0), A371768 (row sums).
%K A371685 nonn,tabl
%O A371685 0,5
%A A371685 _Peter Luschny_, Apr 06 2024