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A339102 Number of compositions (ordered partitions) of n into distinct parts >= 4.

%I A339102 #6 Nov 23 2020 21:01:33
%S A339102 1,0,0,0,1,1,1,1,1,3,3,5,5,7,7,15,15,23,29,37,43,57,87,101,137,175,
%T A339102 235,279,363,431,665,757,1015,1257,1683,2069,2645,3199,4063,5607,6759,
%U A339102 8591,10877,13837,17251,22185,26871,33773,41273,56047,66499,85647,104811
%N A339102 Number of compositions (ordered partitions) of n into distinct parts >= 4.
%H A339102 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339102 G.f.: Sum_{k>=0} k! * x^(k*(k + 7)/2) / Product_{j=1..k} (1 - x^j).
%e A339102 a(9) = 3 because we have [9], [5, 4] and [4, 5].
%p A339102 b:= proc(n, i, p) option remember;
%p A339102       `if`(n=0, p!, `if`((i-3)*(i+4)/2<n, 0,
%p A339102        add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339102     end:
%p A339102 a:= n-> b(n$2, 0):
%p A339102 seq(a(n), n=0..60);  # _Alois P. Heinz_, Nov 23 2020
%t A339102 nmax = 52; CoefficientList[Series[Sum[k! x^(k (k + 7)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A339102 Cf. A008484, A017898, A025149, A032020, A032022, A339101, A339103, A339104.
%K A339102 nonn
%O A339102 0,10
%A A339102 _Ilya Gutkovskiy_, Nov 23 2020