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A339162 Number of compositions (ordered partitions) of n into distinct parts, the least being 1.

%I A339162 #6 Nov 25 2020 20:07:13
%S A339162 0,1,0,2,2,2,8,8,14,14,44,44,74,98,128,272,326,470,644,932,1106,2234,
%T A339162 2552,3800,4958,7070,9068,12140,20042,24674,34256,45632,61814,80630,
%U A339162 109316,135572,217778,262298,362744,466664,636494,805454,1085804,1375388,1776938,2591762
%N A339162 Number of compositions (ordered partitions) of n into distinct parts, the least being 1.
%H A339162 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339162 G.f.: Sum_{k>=1} k! * x^(k*(k + 1)/2) / Product_{j=1..k-1} (1 - x^j).
%e A339162 a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].
%p A339162 b:= proc(n, i, p) option remember;
%p A339162       `if`(n=0, p!, `if`((i-1)*(i+2)/2<n, 0,
%p A339162        add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339162     end:
%p A339162 a:= n-> `if`(n<1, 0, b(n-1$2, 1)):
%p A339162 seq(a(n), n=0..55);  # _Alois P. Heinz_, Nov 25 2020
%t A339162 nmax = 45; CoefficientList[Series[Sum[k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
%Y A339162 Cf. A032020, A096765, A339163, A339164, A339165, A339166.
%K A339162 nonn
%O A339162 0,4
%A A339162 _Ilya Gutkovskiy_, Nov 25 2020