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A339103 Number of compositions (ordered partitions) of n into distinct parts >= 5.

%I A339103 #6 Nov 23 2020 21:14:01
%S A339103 1,0,0,0,0,1,1,1,1,1,1,3,3,5,5,7,7,9,15,17,23,31,37,45,57,65,101,115,
%T A339103 151,189,255,293,383,451,565,777,921,1157,1469,1855,2311,2865,3495,
%U A339103 4313,5231,7063,8269,10509,12849,16217,19829,25171,30031,37485,45183
%N A339103 Number of compositions (ordered partitions) of n into distinct parts >= 5.
%H A339103 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339103 G.f.: Sum_{k>=0} k! * x^(k*(k + 9)/2) / Product_{j=1..k} (1 - x^j).
%e A339103 a(11) = 3 because we have [11], [6, 5] and [5, 6].
%p A339103 b:= proc(n, i, p) option remember;
%p A339103       `if`(n=0, p!, `if`((i-4)*(i+5)/2<n, 0,
%p A339103        add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339103     end:
%p A339103 a:= n-> b(n$2, 0):
%p A339103 seq(a(n), n=0..60);  # _Alois P. Heinz_, Nov 23 2020
%t A339103 nmax = 54; CoefficientList[Series[Sum[k! x^(k (k + 9)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A339103 Cf. A017899, A025150, A032020, A032022, A185325, A339101, A339102, A339104.
%K A339103 nonn
%O A339103 0,12
%A A339103 _Ilya Gutkovskiy_, Nov 23 2020