A127954 Triangle, A097805 * A127648.

1, 0, 2, 0, 2, 3, 0, 2, 6, 4, 0, 2, 9, 12, 5, 0, 2, 12, 24, 20, 6, 0, 2, 15, 40, 50, 30, 7, 0, 2, 18, 60, 100, 90, 42, 8, 0, 2, 21, 84, 175, 210, 147, 56, 9, 0, 2, 24, 112, 280, 420, 392, 224, 72, 10

Offset

1,3

Comments

Row sums = A045623 starting (1, 2, 5, 12, 28, 64, ...).

Dropping the first column gives A128710. - Peter Bala, Mar 05 2013

T(n,k) is the number of ways to place n unlabeled balls into 2 boxes, make compositions of the integer number of balls in each box so that the total number of parts in both compositions is k. - Geoffrey Critzer, Sep 21 2013

Links

Table of n, a(n) for n=1..55.

Formula

A097805 * A127648 as infinite lower triangular matrices.

E.g.f.: 1/(1 - y*(x/(1-x)))^2. - Geoffrey Critzer, Sep 21 2013

O.g.f.: (1+A001263(x,y))^2, - Vladimir Kruchinin, Oct 15 2020

Example

First few rows of the triangle are:
  1;
  0, 2;
  0, 2, 3;
  0, 2, 6, 4;
  0, 2, 9, 12, 5;
  0, 2, 12, 24, 20, 6;
  0, 2, 15, 40, 50, 30, 7;
  ...
T(4,3)=12. Place 4 unlabeled balls into 2 labeled boxes then make compositions of the integer number of balls in each box so that there are a total of 3 parts.
/**** 3 ways since there are 3 compositions of 4 into 3 parts.
*/***  2 ways 1;1+2  and 1;2+1
**/**  2 ways 2;1+1  and 1+1;2.
***/*  2 ways as above.
****/  3 ways as above.
3+2+2+2+3=12. - Geoffrey Critzer, Sep 21 2013

Mathematica

nn=10; a=x/(1-x); CoefficientList[Series[1/(1-y a)^2, {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, Sep 21 2013 *)

Crossrefs

Cf. A001263, A097805, A127648, A128710.

Sequence in context: A198632 A060155 A209127 * A198061 A265583 A339754

Adjacent sequences: A127951 A127952 A127953 * A127955 A127956 A127957

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 09 2007

Status

approved