A117408 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts in which the largest part occurs k times (1<=k<=n).

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 1, 5, 1, 1, 0, 0, 0, 0, 0, 1, 6, 2, 1, 0, 0, 0, 0, 0, 0, 1, 8, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 10, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 12, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Emeric Deutsch, Mar 13 2006

Keywords

• nonn
• tabl

Comments

Row sums yield A000009. T(n,1)=A117409(n). Sum(k*T(n,k),k=1..n)=A092311(n).

Examples

			T(14,2)=4 because we have [7,7],[5,5,3,1],[5,5,1,1,1,1] and [3,3,1,1,1,1,1,1,1,1].
		

Crossrefs

Cf. A000009, A117409, A092311.

Programs

Maple

g:=sum(t*x^(2*k-1)/(1-t*x^(2*k-1))/product(1-x^(2*i-1),i=1..k-1),k=1..40): gser:=simplify(series(g,x=0,35)): for n from 1 to 15 do P[n]:=expand(coeff(gser,x^n)) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form

Formula

G.f.=G(t,x)=sum(tx^(2k-1)/[(1-tx^(2k-1))product(1-x^(2i-1), i=1..k-1)], k=1..infinity).