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%I #9 Feb 13 2020 11:17:56
%S 0,1,1,1,1,1,7,7,13,19,25,31,43,49,61,193,205,337,475,727,985,1363,
%T 1741,2359,2983,3841,4705,5929,12193,13777,20527,27631,39901,52651,
%U 75601,99151,132907,172297,227053,287569,373525,465241,587563,725839,899761,1457683
%N Number of compositions (ordered partitions) of n into distinct parts such that number of parts is odd.
%H Alois P. Heinz, <a href="/A332304/b332304.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: Sum_{k>=1} (2*k - 1)! * x^(k*(2*k - 1)) / Product_{j=1..2*k-1} (1 - x^j).
%F a(n) = A032020(n) - A332305(n).
%e a(6) = 7 because we have [6], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].
%p b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,
%p irem(p, 2)*p!, add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..55); # _Alois P. Heinz_, Feb 09 2020
%t nmax = 45; CoefficientList[Series[Sum[(2 k - 1)! x^(k (2 k - 1))/Product[1 - x^j, {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A027193, A032020, A067659, A166444, A332305.
%K nonn
%O 0,7
%A _Ilya Gutkovskiy_, Feb 09 2020
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