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A339109 Number of compositions (ordered partitions) of n into distinct parts >= 8.

%I A339109 #6 Nov 23 2020 21:50:24
%S A339109 1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3,5,5,7,7,9,9,11,11,19,19,27,33,
%T A339109 41,47,61,67,81,93,107,143,163,199,243,309,353,443,517,631,729,873,
%U A339109 995,1307,1459,1795,2115,2625,3089,3767,4405,5371,6297,7557,8771,10463,12811,14911
%N A339109 Number of compositions (ordered partitions) of n into distinct parts >= 8.
%H A339109 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339109 G.f.: Sum_{k>=0} k! * x^(k*(k + 15)/2) / Product_{j=1..k} (1 - x^j).
%e A339109 a(17) = 3 because we have [17], [9, 8] and [8, 9].
%p A339109 b:= proc(n, i, p) option remember;
%p A339109       `if`(n=0, p!, `if`((i-7)*(i+8)/2<n, 0,
%p A339109        add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339109     end:
%p A339109 a:= n-> b(n$2, 0):
%p A339109 seq(a(n), n=0..64);  # _Alois P. Heinz_, Nov 23 2020
%t A339109 nmax = 64; CoefficientList[Series[Sum[k! x^(k (k + 15)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A339109 Cf. A017902, A025153, A032020, A032022, A185328, A339101, A339102, A339103, A339104, A339108, A339110.
%K A339109 nonn
%O A339109 0,18
%A A339109 _Ilya Gutkovskiy_, Nov 23 2020