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A212431 Triangle read by rows: row sums, right and left borders are the Bell sequence, or a shifted variant. See Comments for precise definition.

%I A212431 #20 Jun 12 2018 04:23:43
%S A212431 1,1,1,2,1,2,5,3,2,5,15,9,8,5,15,52,31,28,25,15,52,203,121,108,100,90,
%T A212431 52,203,877,523,466,425,405,364,203,877,4140,2469,2202,2000,1875,1820,
%U A212431 1624,877,4140,21147,12611,11250,10230,9525,9100,8932,7893,4140,21147
%N A212431 Triangle read by rows: row sums, right and left borders are the Bell sequence, or a shifted variant. See Comments for precise definition.
%C A212431 Consider A186020 as an infinite lower triangular matrix, and multiply the columns successively by the Bell numbers A000110, (1, 1, 2, 5, 15, 52,...).
%C A212431 Right and left borders = the Bell numbers, A000110: (1, 1, 2, 5, 15,...). Row sums = the shifted Bell numbers, (1, 2, 5, 15, 52,...).
%C A212431 Sum of n-th row terms = rightmost and leftmost terms of next row.
%H A212431 Alois P. Heinz, <a href="/A212431/b212431.txt">Rows n = 0..140, flattened</a>
%e A212431 First few rows of the triangle are:
%e A212431 1;
%e A212431 1, 1
%e A212431 2, 1, 2;
%e A212431 5, 3, 2, 5;
%e A212431 15, 9, 8, 5, 15;
%e A212431 52, 31, 28, 25, 15, 52;
%e A212431 203, 121, 108, 100, 90, 52, 203;
%e A212431 877, 523, 466, 425, 405, 364, 203, 877;
%e A212431 4140, 2469, 2202, 2000, 1875, 1820, 1624, 877, 4140;
%e A212431 21147, 12611, 11250, 10230, 9525, 9100, 8932, 7893, 4140, 21147;
%e A212431 ...
%p A212431 b:= proc(n) option remember; `if`(n=0, [1, 0],
%p A212431       add((p-> p+[0, p[1]*x^(n-j)])(b(n-j)*
%p A212431       binomial(n-1, j-1)), j=1..n))
%p A212431     end:
%p A212431 T:= n-> (p-> seq(`if`(i=n, p[1], coeff(
%p A212431          p[2], x, i)), i=0..n))(b(n)):
%p A212431 seq(T(n), n=0..12);  # _Alois P. Heinz_, May 16 2017
%t A212431 b[n_] := b[n] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, p[[1]]*x^(n - j)}][b[n - j]*Binomial[n - 1, j - 1]], {j, 1, n}]];
%t A212431 T[n_] := Function[p, Table[If[i == n, p[[1]], Coefficient[p[[2]], x, i]], {i, 0, n}]][b[n]];
%t A212431 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jun 12 2018, after _Alois P. Heinz_ *)
%Y A212431 Cf. A000110, A186020.
%K A212431 nonn,tabl
%O A212431 0,4
%A A212431 _Gary W. Adamson_, Jun 21 2012
%E A212431 Edited by _N. J. A. Sloane_, Jun 22 2012