A352366 Triangle read by rows. The incomplete Bell transform of the Catalan numbers.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 5, 11, 6, 1, 0, 14, 45, 35, 10, 1, 0, 42, 199, 210, 85, 15, 1, 0, 132, 938, 1309, 700, 175, 21, 1, 0, 429, 4675, 8498, 5789, 1890, 322, 28, 1, 0, 1430, 24489, 57455, 48762, 19929, 4410, 546, 36, 1

Offset: 0

Author

Peter Luschny, Mar 15 2022

Keywords

• nonn
• tabl

Examples

			Triangle start:
[0] 1;
[1] 0,    1;
[2] 0,    1,     1;
[3] 0,    2,     3,     1;
[4] 0,    5,    11,     6,     1;
[5] 0,   14,    45,    35,    10,     1;
[6] 0,   42,   199,   210,    85,    15,    1;
[7] 0,  132,   938,  1309,   700,   175,   21,   1;
[8] 0,  429,  4675,  8498,  5789,  1890,  322,  28,  1;
[9] 0, 1430, 24489, 57455, 48762, 19929, 4410, 546, 36, 1;
		

Crossrefs

Cf. A000108, A352367 (row sums), A352368 (alternating row sums).

Cf. A352363, A352369.

Programs

Maple

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
for n from 0 to 9 do
seq(IncompleteBellB(n, k, seq(CatalanNumber(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) = CatalanNumber(n).