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%I #5 Jun 20 2018 01:28:54
%S 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,
%T 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,
%U 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
%N 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.
%C Row sums yield A000929. sum(k*T(n,k),k=1..n)=A117146(n).
%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 G.f.: G(t,x) = -1+1/product(1-tx^(2^k-1), k=1..infinity).
%e T(9,3)=2 because we have [7,1,1] and [3,3,3].
%p 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
%Y Cf. A000929, A117146.
%K nonn,tabl
%O 1,39
%A _Emeric Deutsch_, Mar 06 2006
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