A116799 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts such that the largest part is k (n>=1, k>=1).

1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 0, 2, 0, 1, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 0, 3, 0, 3, 0, 2, 0, 1, 1, 0, 3, 0, 4, 0, 2, 0, 1, 0, 1, 1, 0, 4, 0, 4, 0, 3, 0, 2, 0, 1, 1, 0, 4, 0, 5, 0, 4, 0, 2, 0, 1, 0, 1, 1, 0, 4, 0, 6, 0, 5, 0, 3, 0, 2, 0, 1, 1, 0, 5, 0, 7, 0, 6

Offset: 1

Author

Emeric Deutsch, Feb 24 2006

Keywords

• nonn
• tabf

Comments

Both rows 2n-1 and 2n have 2n-1 terms. Row sums yield A000009. T(n,2k)=0. T(n,3)=A002264(n). Sum(k*T(n,k),k>=1)=A092322(n).

Examples

			T(10,5)=3 because we have [3,3,3,1], [3,3,1,1,1,1] and [3,1,1,1,1,1,1,1].
Triangle starts:
1;
1;
1,0,1;
1,0,1;
1,0,1,0,1;
1,0,2,0,1;
		

Crossrefs

Cf. A000009, A002264, A092322, A116856.

Programs

Maple

g:=sum(t^(2*j-1)*x^(2*j-1)/product(1-x^(2*i-1),i=1..j),j=1..40): gser:=simplify(series(g,x=0,22)): for n from 1 to 16 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 16 do seq(coeff(P[n],t^j),j=1..2*ceil(n/2)-1) od; # yields sequence in triangular form

Formula

G.f.=sum(t^(2j-1)*x^(2j-1)/product(1-x^(2i-1), i=1..j), j=1..infinity).