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

%I A357583 #10 Apr 06 2025 14:53:35
%S A357583 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,
%T A357583 0,877,764,471,212,65,12,1,0,4140,3311,2013,974,355,90,14,1,0,21147,
%U A357583 15378,8951,4440,1790,550,119,16,1,0,115975,76418,41745,20526,8727,3027,805,152,18,1
%N A357583 Triangle read by rows. Convolution triangle of the Bell numbers.
%F A357583 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
%e A357583 Triangle T(n, k) starts:
%e A357583   [0] 1;
%e A357583   [1] 0,     1;
%e A357583   [2] 0,     2,     1;
%e A357583   [3] 0,     5,     4,    1;
%e A357583   [4] 0,    15,    14,    6,    1;
%e A357583   [5] 0,    52,    50,   27,    8,    1;
%e A357583   [6] 0,   203,   189,  113,   44,   10,   1;
%e A357583   [7] 0,   877,   764,  471,  212,   65,  12,   1;
%e A357583   [8] 0,  4140,  3311, 2013,  974,  355,  90,  14,  1;
%e A357583   [9] 0, 21147, 15378, 8951, 4440, 1790, 550, 119, 16, 1;
%p A357583 # Using function PMatrix from A357368.
%p A357583 PMatrix(10, combinat[bell]);
%Y A357583 Cf. A000110, A129247 (row sums), A007311, A357584 (central terms).
%K A357583 nonn,tabl
%O A357583 0,5
%A A357583 _Peter Luschny_, Oct 05 2022