A099040 Riordan array (1, 2+2x).

1, 0, 2, 0, 2, 4, 0, 0, 8, 8, 0, 0, 4, 24, 16, 0, 0, 0, 24, 64, 32, 0, 0, 0, 8, 96, 160, 64, 0, 0, 0, 0, 64, 320, 384, 128, 0, 0, 0, 0, 16, 320, 960, 896, 256, 0, 0, 0, 0, 0, 160, 1280, 2688, 2048, 512, 0, 0, 0, 0, 0, 32, 960, 4480, 7168, 4608, 1024, 0, 0, 0, 0, 0, 0, 384, 4480, 14336, 18432, 10240, 2048

Offset: 0

Author

Paul Barry, Sep 23 2004

Keywords

• easy
• nonn
• tabl

Comments

Row sums give A002605. Diagonal sums give A052907.

The Riordan array (1,s+t*x) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

T(n,k) is the number of compositions of n into two types of parts of size 1 and 2 that have exactly k parts. - Geoffrey Critzer, Aug 18 2012.

Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 22 2020

Examples

			Rows begin {1}, {0,2}, {0,2,4}, {0,0,8,8}, {0,0,4,24,16}, {0,0,0,24,64,32},...
T(3,2)=8 because we have: 1+2,1+2',1'+2,1'+2',2+1,2+1',2'+1,2'+1' where a part of the second type is designated by '. - _Geoffrey Critzer_, Aug 18 2012
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A002605, A026729, A038208, A052907.

Programs

Mathematica

nn = 8; CoefficientList[Series[1/(1 - 2 y x - 2 y x^2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 18 2012 *)

Formula

Number triangle T(n, k) = 2^k*binomial(k, n-k).

Columns have g.f. (2x+2x^2)^k.

T(n,k) = A026729(n,k)*2^k. - Philippe Deléham, Jul 28 2006

O.g.f.: 1/(1-2*y*x-2*y*x^2). - Geoffrey Critzer, Aug 18 2012.