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%I #32 Nov 22 2016 13:56:47
%S 1,1,0,1,2,1,2,1,4,7,4,7,6,13,6,19,32,25,32,31,58,43,82,49,132,181,
%T 156,193,230,325,278,457,376,715,448,967,1290,1345,1386,1723,2276,
%U 2341,3116,2959,4750,3823,6358,4681,9480,10945,11832,12169,16442,18793,21002,25537,27820,37687
%N Number of compositions (ordered partitions) of n into distinct odd parts.
%H Alois P. Heinz, <a href="/A032021/b032021.txt">Table of n, a(n) for n = 0..10000</a>
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F "AGK" (ordered, elements, unlabeled) transform of 1, 0, 1, 0...(odds)
%F G.f.: sum(k>=0, k! * x^(k^2) / prod(j=1..k, 1-x^(2*j) ) ). - _Vladeta Jovovic_, Aug 05 2004
%p b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
%p ->x+y, b(n, i-2), [0, `if`(i>n, [], b(n-i, i-2))[]], 0)))
%p end:
%p a:= proc(n) local l; l:= b(n, n-1+irem(n, 2));
%p a(n):= add(l[i]*(i-1)!, i=1..nops(l))
%p end:
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Nov 09 2012
%t b[n_, i_] := If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-2], Join[{0}, If[i>n, {}, b[n-i, i-2]]]}]]]; a[n_] := Module[{l}, l = b[n, n-1+Mod[n, 2]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)
%o (PARI)
%o N=66; q='q+O('q^N);
%o gf=sum(k=0,N, k! * q^(k^2) / prod(j=1,k, 1-q^(2*j) ) );
%o Vec(gf)
%o /* _Joerg Arndt_, Sep 17 2012 */
%Y Cf. A032020, A000700.
%K nonn
%O 0,5
%A _Christian G. Bower_
%E Prepended a(0)=1, _Joerg Arndt_, Oct 20 2012
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