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

0,4

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

Links

Table of n, a(n) for n=0..54.

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

Formula

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

Example

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;

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 *)

Crossrefs

First column are the Bell numbers A000110.

Row sums are A154381, alternating row sums are A000007.

Cf. A321620.

Sequence in context: A147703 A147747 A039599 * A155083 A011357 A080409

Adjacent sequences: A154377 A154378 A154379 * A154381 A154382 A154383

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 08 2009

Extensions

New name by Peter Luschny, Dec 06 2018

Status

approved