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%I #8 Jun 20 2018 01:28:48
%S 1,2,4,6,8,12,16,20,27,34,40,50,60,70,85,100,115,136,156,176,206,234,
%T 261,300,336,370,418,466,511,572,633,690,765,840,914,1008,1102,1194,
%U 1307,1420,1530,1668,1806,1940,2107,2272,2431,2626,2825,3016,3246,3484
%N 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,...).
%H P. C. P. Bhatt, <a href="https://doi.org/10.1016/S0020-0190(99)00090-3">An interesting way to partition a number</a>, Inform. Process. Lett., 71, 1999, 141-148.
%H W. M. Y. Goh, P. Hitczenko and A. Shokoufandeh, <a href="https://doi.org/10.1016/S0020-0190(01)00300-3">s-partitions</a>, Inform. Process. Lett., 82, 2002, 327-329.
%F a(n) = sum(k*A117145(n,k), k=1..n).
%F G.f.: sum(x^(2^k-1)/(1-x^(2^k-1)), k=1..infinity)/product(1-x^(2^k-1), k=1..infinity).
%e 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.
%p 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);
%Y Cf. A117145.
%K nonn
%O 0,2
%A _Emeric Deutsch_, Mar 06 2006
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