A357583 Triangle read by rows. Convolution triangle of the Bell numbers.

1, 0, 1, 0, 2, 1, 0, 5, 4, 1, 0, 15, 14, 6, 1, 0, 52, 50, 27, 8, 1, 0, 203, 189, 113, 44, 10, 1, 0, 877, 764, 471, 212, 65, 12, 1, 0, 4140, 3311, 2013, 974, 355, 90, 14, 1, 0, 21147, 15378, 8951, 4440, 1790, 550, 119, 16, 1, 0, 115975, 76418, 41745, 20526, 8727, 3027, 805, 152, 18, 1

Offset: 0

Author

Peter Luschny, Oct 05 2022

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0,     1;
  [2] 0,     2,     1;
  [3] 0,     5,     4,    1;
  [4] 0,    15,    14,    6,    1;
  [5] 0,    52,    50,   27,    8,    1;
  [6] 0,   203,   189,  113,   44,   10,   1;
  [7] 0,   877,   764,  471,  212,   65,  12,   1;
  [8] 0,  4140,  3311, 2013,  974,  355,  90,  14,  1;
  [9] 0, 21147, 15378, 8951, 4440, 1790, 550, 119, 16, 1;
		

Crossrefs

Cf. A000110, A129247 (row sums), A007311, A357584 (central terms).

Programs

Maple

# Using function PMatrix from A357368.
PMatrix(10, combinat[bell]);

Formula

Conjecture: row polynomials are x*R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) + x*R(n-1,1)*R(1,k) for n > 1, k > 0 with R(1,k) = Bell(k) for k > 0. The same recursion seems to work for self-convolution of any other sequence. - Mikhail Kurkov, Apr 05 2025