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A339163 Number of compositions (ordered partitions) of n into distinct parts, the least being 2.

%I A339163 #6 Nov 25 2020 21:28:20
%S A339163 0,0,1,0,0,2,2,2,2,8,8,14,14,20,44,50,74,104,128,158,326,356,524,698,
%T A339163 986,1160,1592,2606,3158,4316,5708,7706,10082,12920,16136,25718,30614,
%U A339163 41756,53396,71978,91058,122144,149384,193670,279614,342860,447764,581234
%N A339163 Number of compositions (ordered partitions) of n into distinct parts, the least being 2.
%H A339163 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A339163 G.f.: Sum_{k>=1} k! * x^(k*(k + 3)/2) / Product_{j=1..k-1} (1 - x^j).
%e A339163 a(9) = 8 because we have [7, 2], [4, 3, 2], [4, 2, 3], [3, 4, 2], [3, 2, 4], [2, 7], [2, 4, 3] and [2, 3, 4].
%p A339163 b:= proc(n, i, p) option remember;
%p A339163       `if`(n=0, p!, `if`((i-2)*(i+3)/2<n, 0,
%p A339163        add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
%p A339163     end:
%p A339163 a:= n-> `if`(n<2, 0, b(n-2$2, 1)):
%p A339163 seq(a(n), n=0..55);  # _Alois P. Heinz_, Nov 25 2020
%t A339163 nmax = 47; CoefficientList[Series[Sum[k! x^(k (k + 3)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
%Y A339163 Cf. A002865, A032022, A096749, A339162, A339164, A339165, A339166.
%K A339163 nonn
%O A339163 0,6
%A A339163 _Ilya Gutkovskiy_, Nov 25 2020