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
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;
Programs
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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
Comments