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A339108 Number of compositions (ordered partitions) of n into distinct parts >= 7.

%I A339108 #8 Dec 07 2020 14:52:01
%S A339108 1,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3,5,5,7,7,9,9,11,17,19,25,33,39,47,
%T A339108 59,67,79,93,129,143,185,223,289,333,423,491,611,703,847,1089,1281,
%U A339108 1547,1889,2323,2833,3417,4095,4967,5939,7099,8359,10653,12345,15047,17993
%N A339108 Number of compositions (ordered partitions) of n into distinct parts >= 7.
%H A339108 Alois P. Heinz, <a href="/A339108/b339108.txt">Table of n, a(n) for n = 0..5000</a>
%H A339108 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339108 G.f.: Sum_{k>=0} k! * x^(k*(k + 13)/2) / Product_{j=1..k} (1 - x^j).
%e A339108 a(15) = 3 because we have [15], [8, 7] and [7, 8].
%p A339108 b:= proc(n, i, p) option remember;
%p A339108       `if`(n=0, p!, `if`((i-6)*(i+7)/2<n, 0,
%p A339108        add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339108     end:
%p A339108 a:= n-> b(n$2, 0):
%p A339108 seq(a(n), n=0..64);  # _Alois P. Heinz_, Nov 23 2020
%t A339108 nmax = 60; CoefficientList[Series[Sum[k! x^(k (k + 13)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A339108 Cf. A017901, A025152, A032020, A032022, A185327, A339101, A339102, A339103, A339104, A339109, A339110.
%K A339108 nonn
%O A339108 0,16
%A A339108 _Ilya Gutkovskiy_, Nov 23 2020