A117145 Triangle read by rows: T(n,k) is the number of partitions of n into parts of the form 2^j-1, j=1,2,... and having k parts (n>=1, k>=1). Partitions into parts of the form 2^j-1, j=1,2,... are called s-partitions.

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1

Offset: 1

Author

Emeric Deutsch, Mar 06 2006

Keywords

• nonn
• tabl

Comments

Row sums yield A000929. sum(k*T(n,k),k=1..n)=A117146(n).

Examples

			T(9,3)=2 because we have [7,1,1] and [3,3,3].
		

Links

• P. C. P. Bhatt, An interesting way to partition a number, Inform. Process. Lett., 71, 1999, 141-148.
• W. M. Y. Goh, P. Hitczenko and A. Shokoufandeh, s-partitions, Inform. Process. Lett., 82, 2002, 327-329.

Crossrefs

Cf. A000929, A117146.

Programs

Maple

g:=-1+1/product(1-t*x^(2^k-1),k=1..10): gser:=simplify(series(g,x=0,20)): for n from 1 to 19 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 19 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form

Formula

G.f.: G(t,x) = -1+1/product(1-tx^(2^k-1), k=1..infinity).