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A339164 Number of compositions (ordered partitions) of n into distinct parts, the least being 3.

%I A339164 #6 Nov 25 2020 21:21:31
%S A339164 0,0,0,1,0,0,0,2,2,2,2,2,8,8,14,14,20,20,50,50,80,104,134,158,212,356,
%T A339164 410,578,752,1040,1238,1646,1964,3236,3674,5066,6368,8720,10862,14078,
%U A339164 17180,22076,31802,38378,49784,63824,82670,104150,136220,165980
%N A339164 Number of compositions (ordered partitions) of n into distinct parts, the least being 3.
%H A339164 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339164 G.f.: Sum_{k>=1} k! * x^(k*(k + 5)/2) / Product_{j=1..k-1} (1 - x^j).
%e A339164 a(12) = 8 because we have [9, 3], [5, 4, 3], [5, 3, 4], [4, 5, 3], [4, 3, 5], [3, 9], [3, 5, 4] and [3, 4, 5].
%p A339164 b:= proc(n, i, p) option remember;
%p A339164       `if`(n=0, p!, `if`((i-3)*(i+4)/2<n, 0,
%p A339164        add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339164     end:
%p A339164 a:= n-> `if`(n<3, 0, b(n-3$2, 1)):
%p A339164 seq(a(n), n=0..55);  # _Alois P. Heinz_, Nov 25 2020
%t A339164 nmax = 49; CoefficientList[Series[Sum[k! x^(k (k + 5)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
%Y A339164 Cf. A026796, A026824, A339101, A339162, A339163, A339165, A339166.
%K A339164 nonn
%O A339164 0,8
%A A339164 _Ilya Gutkovskiy_, Nov 25 2020