A154380 The Riordan square of the Bell numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 15, 29, 20, 7, 1, 52, 102, 77, 35, 9, 1, 203, 392, 302, 157, 54, 11, 1, 877, 1641, 1235, 683, 277, 77, 13, 1, 4140, 7451, 5324, 2987, 1329, 445, 104, 15, 1, 21147, 36525, 24329, 13391, 6230, 2340, 669, 135, 17, 1

Offset: 0

Author

Paul Barry, Jan 08 2009

Keywords

• easy
• nonn
• tabl

Comments

The Riordan square is defined in A321620.

Previous name was: Triangle T(n,k), 0<=k<=n, read by rows given by [1, 1, 1, 2, 1, 3, 1, 4, 1, ...] DELTA [1, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

In general, the triangle [r_0, r_1, r_2, ...] DELTA [s_0, s_1, s_2, ...] has generating function

1/(1 - (r_0*x + s_0*x*y)/(1 - (r_1*x + s_1*x*y)/(1 - (r_2*x + s_2*x*y)/(1 -... (continued fraction)

A130167*A007318 as infinite lower triangular matrices. - Philippe Deléham, Jan 11 2009

Examples

			Triangle begins
     1;
     1,   1;
     2,   3,   1;
     5,   9,   5,   1;
    15,  29,  20,   7,  1;
    52, 102,  77,  35,  9,  1;
   203, 392, 302, 157, 54, 11, 1;
		

Links

• Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.

Crossrefs

First column are the Bell numbers A000110.

Row sums are A154381, alternating row sums are A000007.

Cf. A321620.

Programs

Maple

# The function RiordanSquare is defined in A321620.
RiordanSquare(add(x^k/mul(1-j*x, j=1..k), k=0..10), 10); # Peter Luschny, Dec 06 2018

Mathematica

RiordanSquare[gf_, len_] := Module[{T}, T[n_, k_] := T[n, k] = If[k == 0, SeriesCoefficient[gf, {x, 0, n}], Sum[T[j, k - 1] T[n - j, 0], {j, k - 1, n - 1}]]; Table[T[n, k], {n, 0, len - 1}, {k, 0, n}]];
RiordanSquare[Sum[x^k/Product[1 - j x, {j, 1, k}], {k, 0, 10}], 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

Formula

G.f.: 1/(1-(x+xy)/(1-x/(1-x/(1-2x/(1-x/(1-3x/(1-x/(1-4x/(1-... (continued fraction).

Extensions

New name by Peter Luschny, Dec 06 2018