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A032022 Number of compositions (ordered partitions) of n into distinct parts >= 2.

%I A032022 #27 Sep 20 2019 03:17:53
%S A032022 1,0,1,1,1,3,3,5,5,13,13,21,27,35,65,79,109,147,207,245,449,517,745,
%T A032022 957,1335,1691,2237,3463,4273,5787,7611,10109,13061,17413,21493,32853,
%U A032022 39627,53675,68321,91663,114997,152811,192063,245885,346649,428869,557305
%N A032022 Number of compositions (ordered partitions) of n into distinct parts >= 2.
%H A032022 Alois P. Heinz, <a href="/A032022/b032022.txt">Table of n, a(n) for n = 0..1000</a>
%H A032022 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A032022 "AGK" (ordered, elements, unlabeled) transform of 0, 1, 1, 1...
%F A032022 G.f.: sum(i>=0, i! * x^((i^2+3*i)/2) / prod(j=1..i, 1-x^j ) ). - _Vladeta Jovovic_, May 21 2006
%p A032022 b:= proc(n, i) option remember; local s; s:= i*(i+1)/2-1;
%p A032022      `if`(n=0, [1], `if`(i<2 or n>s, [], zip((x, y)->x+y,
%p A032022       b(n, i-1), [0, `if`(i>n, [], b(n-i, i-1))[]], 0)))
%p A032022     end:
%p A032022 a:= proc(n) option remember; local l; l:= b(n$2);
%p A032022       add(l[i]*(i-1)!, i=1..nops(l))
%p A032022     end:
%p A032022 seq(a(n), n=0..70);  # _Alois P. Heinz_, Nov 09 2012
%t A032022 zip = With[{m = Max[Length[#1], Length[#2]]}, PadRight[#1, m] + PadRight[#2, m]]&; b[n_, i_] := b[n, i] = With[{s = i*(i+1)/2-1}, If[n == 0, {1}, If[i<2 || n>s, {}, zip[ b[n, i-1], Join[{0}, If[i>n, {}, b[n-i, i - 1]]]]]]]; a[n_] := a[n] = Module[{l = b[n, n]}, Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]]; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Feb 13 2017, after _Alois P. Heinz_ *)
%Y A032022 Cf. A032020.
%K A032022 nonn
%O A032022 0,6
%A A032022 _Christian G. Bower_
%E A032022 Prepended a(0)=1, _Joerg Arndt_, Oct 20 2012