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

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, 52, 203, 877, 523, 466, 425, 405, 364, 203, 877, 4140, 2469, 2202, 2000, 1875, 1820, 1624, 877, 4140, 21147, 12611, 11250, 10230, 9525, 9100, 8932, 7893, 4140, 21147

Offset: 0

Author

Gary W. Adamson, Jun 21 2012

Keywords

• nonn
• tabl

Comments

Consider A186020 as an infinite lower triangular matrix, and multiply the columns successively by the Bell numbers A000110, (1, 1, 2, 5, 15, 52,...).

Right and left borders = the Bell numbers, A000110: (1, 1, 2, 5, 15,...). Row sums = the shifted Bell numbers, (1, 2, 5, 15, 52,...).

Sum of n-th row terms = rightmost and leftmost terms of next row.

Examples

			First few rows of the triangle are:
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, 52, 203;
877, 523, 466, 425, 405, 364, 203, 877;
4140, 2469, 2202, 2000, 1875, 1820, 1624, 877, 4140;
21147, 12611, 11250, 10230, 9525, 9100, 8932, 7893, 4140, 21147;
...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A000110, A186020.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1, 0],
      add((p-> p+[0, p[1]*x^(n-j)])(b(n-j)*
      binomial(n-1, j-1)), j=1..n))
    end:
T:= n-> (p-> seq(`if`(i=n, p[1], coeff(
         p[2], x, i)), i=0..n))(b(n)):
seq(T(n), n=0..12);  # Alois P. Heinz, May 16 2017

Mathematica

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[n_] := Function[p, Table[If[i == n, p[[1]], Coefficient[p[[2]], x, i]], {i, 0, n}]][b[n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 12 2018, after Alois P. Heinz *)

Extensions

Edited by N. J. A. Sloane, Jun 22 2012