A117146 Number of parts in all s-partitions of n. An s-partition of n is a partition of n into parts of the form 2^j-1 (j=1,2,...).

1, 2, 4, 6, 8, 12, 16, 20, 27, 34, 40, 50, 60, 70, 85, 100, 115, 136, 156, 176, 206, 234, 261, 300, 336, 370, 418, 466, 511, 572, 633, 690, 765, 840, 914, 1008, 1102, 1194, 1307, 1420, 1530, 1668, 1806, 1940, 2107, 2272, 2431, 2626, 2825, 3016, 3246, 3484

Offset: 0

Author

Emeric Deutsch, Mar 06 2006

Keywords

• nonn

Examples

			a(7)=16 because the s-partitions of 7 are [7],[3,3,1],[3,1,1,1,1] and [1,1,1,1,1,1,1], with a total of 1+3+5+7=16 parts.
		

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. A117145.

Programs

Maple

g:=sum(x^(2^k-1)/(1-x^(2^k-1)),k=1..10)/product(1-x^(2^k-1),k=1..10): gser:=series(g,x=0,60): seq(coeff(gser,x^n),n=1..56);

Formula

a(n) = sum(k*A117145(n,k), k=1..n).

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