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A339101 Number of compositions (ordered partitions) of n into distinct parts >= 3.

%I A339101 #8 Nov 24 2020 07:14:09
%S A339101 1,0,0,1,1,1,1,3,3,5,5,7,13,15,21,29,35,43,79,87,123,161,221,259,349,
%T A339101 531,645,857,1115,1471,1903,2403,2979,4493,5357,7135,9013,11919,14925,
%U A339101 19685,23939,30667,42679,52215,67035,86009,109541,137923,177493,222027,277749
%N A339101 Number of compositions (ordered partitions) of n into distinct parts >= 3.
%H A339101 Alois P. Heinz, <a href="/A339101/b339101.txt">Table of n, a(n) for n = 0..5000</a>
%H A339101 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339101 G.f.: Sum_{k>=0} k! * x^(k*(k + 5)/2) / Product_{j=1..k} (1 - x^j).
%e A339101 a(7) = 3 because we have [7], [4, 3] and [3, 4].
%p A339101 b:= proc(n, i, p) option remember;
%p A339101       `if`(n=0, p!, `if`((i-2)*(i+3)/2<n, 0, add(
%p A339101          b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339101     end:
%p A339101 a:= n-> b(n$2, 0):
%p A339101 seq(a(n), n=0..50);  # _Alois P. Heinz_, Nov 23 2020
%t A339101 nmax = 50; CoefficientList[Series[Sum[k! x^(k (k + 5)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A339101 Cf. A008483, A025148, A032020, A032022, A078012, A339102, A339103, A339104.
%K A339101 nonn
%O A339101 0,8
%A A339101 _Ilya Gutkovskiy_, Nov 23 2020