A352363 Triangle read by rows. The incomplete Bell transform of the swinging factorials A056040.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 6, 50, 35, 10, 1, 0, 30, 166, 225, 85, 15, 1, 0, 20, 756, 1246, 735, 175, 21, 1, 0, 140, 2932, 7588, 5761, 1960, 322, 28, 1, 0, 70, 11556, 45296, 46116, 20181, 4536, 546, 36, 1

Offset: 0

Author

Peter Luschny, Mar 15 2022

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 0,   1;
[2] 0,   1,     1;
[3] 0,   2,     3,     1;
[4] 0,   6,    11,     6,     1;
[5] 0,   6,    50,    35,    10,     1;
[6] 0,  30,   166,   225,    85,    15,    1;
[7] 0,  20,   756,  1246,   735,   175,   21,   1;
[8] 0, 140,  2932,  7588,  5761,  1960,  322,  28, 1;
[9] 0,  70, 11556, 45296, 46116, 20181, 4536, 546, 36, 1;
		

Crossrefs

Cf. A056040, A352364 (row sums), A352365 (alternating row sums).

Cf. A352366, A352369.

Programs

Maple

SwingNumber := n -> n! / iquo(n, 2)!^2:
for n from 0 to 9 do
seq(IncompleteBellB(n, k, seq(SwingNumber(j), j = 0..n)), k = 0..n) od;

Formula

Given a sequence s let s|n denote the initial segment s(0), s(1), ..., s(n).

(T(s))(n, k) = IncompleteBellPolynomial(n, k, s|n), where s(n) = n!/floor(n/2)!^2.