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
FORMULA
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
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Feb 09 2007
STATUS
approved