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

0,5

Links

Table of n, a(n) for n=0..44.

Formula

Restricted to the range 1 <= k <= n: T(n, k) = n!*(2^k - 1)/k.

Example

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;

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);

Crossrefs

Cf. A029767 (main diagonal), A024167 (column 0), A371768 (row sums).

Sequence in context: A018126 A087900 A101216 * A175095 A073216 A181902

Adjacent sequences: A371682 A371683 A371684 * A371686 A371687 A371688

Keyword

nonn,tabl

Author

Peter Luschny, Apr 06 2024

Status

approved