A116683 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, in which the sum of the even parts is k (n>=0, 0<=k<=n).

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 1, 0, 1, 0, 2, 0, 2, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 3, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 3, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 3, 0, 3, 0, 2, 0, 4

Offset: 0

Author

Emeric Deutsch, Feb 22 2006

Keywords

• nonn
• tabf

Comments

Row 2n-1 has 2n-1 terms; row 2n has 2n+1 terms. Row sums yield A000009. T(n,0)=A000700(n). Columns 2n-1 contain only 0's. Sum(k*T(n,k), k=0..n)=A116684(n).

Examples

			T(9,6)=2 because we have [6,3] and [4,3,2].
Triangle starts:
1;
1;
0,0,1;
1,0,1;
1,0,0,0,1;
1,0,1,0,1;
1,0,1,0,0,0,2
		

Crossrefs

Cf. A000009, A000700, A116684.

Programs

Maple

g:=product((1+x^(2*j-1))*(1+(t*x)^(2*j)),j=1..30): gser:=simplify(series(g,x=0,20)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..2*floor(n/2)) od; # yields sequence in triangular form

Formula

G.f.=product((1+x^(2j-1))(1+(tx)^(2j)), j=1..infinity).