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A339110 Number of compositions (ordered partitions) of n into distinct parts >= 9.

%I A339110 #5 Nov 23 2020 21:51:48
%S A339110 1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,5,5,7,7,9,9,11,11,13,19,21,
%T A339110 27,35,41,49,61,69,81,95,107,121,163,177,219,263,329,373,469,537,657,
%U A339110 755,899,1021,1219,1485,1707,2027,2417,2881,3445,4077,4809,5735,6755,7969,9307
%N A339110 Number of compositions (ordered partitions) of n into distinct parts >= 9.
%H A339110 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339110 G.f.: Sum_{k>=0} k! * x^(k*(k + 17)/2) / Product_{j=1..k} (1 - x^j).
%e A339110 a(19) = 3 because we have [19], [10, 9] and [9, 10].
%p A339110 b:= proc(n, i, p) option remember;
%p A339110       `if`(n=0, p!, `if`((i-8)*(i+9)/2<n, 0,
%p A339110        add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339110     end:
%p A339110 a:= n-> b(n$2, 0):
%p A339110 seq(a(n), n=0..69);  # _Alois P. Heinz_, Nov 23 2020
%t A339110 nmax = 66; CoefficientList[Series[Sum[k! x^(k (k + 17)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A339110 Cf. A017903, A025154, A032020, A032022, A185329, A339101, A339102, A339103, A339104, A339108, A339109.
%K A339110 nonn
%O A339110 0,20
%A A339110 _Ilya Gutkovskiy_, Nov 23 2020