A113686 Triangular array T(n,k)=number of partitions of n in which sum of even parts is k, for k=0,1,...n; n>=0.

1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2, 3, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 3, 5, 0, 3, 0, 4, 0, 3, 0, 6, 0, 4, 0, 4, 0, 3, 0, 5, 8, 0, 5, 0, 6, 0, 6, 0, 5, 0, 10, 0, 6, 0, 8, 0, 6, 0, 5, 0, 7, 12, 0, 8, 0, 10, 0, 9, 0, 10, 0, 7, 0, 15, 0, 10, 0, 12, 0, 12, 0, 10, 0, 7, 0, 11, 18, 0, 12, 0, 16, 0, 15

Offset: 0

Author

Clark Kimberling, Nov 05 2005

Keywords

• nonn
• tabl

Comments

(Sum over row n) = A000041(n) = number of partitions of n. Reversal of this array is array in A113685, except for row 0.

Sum(k*T(n,k),k=0..n)=A066966(n). - Emeric Deutsch, Feb 17 2006

Examples

			First 5 rows:
1
1 0
1 0 1
2 0 1 0
2 0 1 0 2
3 0 2 0 2 0.
The partitions of 5 are
5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1;
sums of even parts are 0,4,2,0,4,2, respectively,
so that the numbers of 0's, 1's, 2s, 3s, 4s, 5s
are 0,3,0,2,0,2,0, which is row 5 of the array.
		

Crossrefs

Cf. A000041, A113685.

Cf. A066966.

Programs

Maple

g:=1/product((1-x^(2*j-1))*(1-t^(2*j)*x^(2*j)),j=1..20): gser:=simplify(series(g,x=0,20)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(gser,x^n) od: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form - Emeric Deutsch, Feb 17 2006

Formula

G:=1/product((1-x^(2j-1))(1-t^(2j)x^(2j)), j=1..infinity). - Emeric Deutsch, Feb 17 2006

Extensions

More terms from Emeric Deutsch, Feb 17 2006