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A216708 Number of compositions (ordered partitions) of n into 2 or more distinct nonnegative parts.

%I A216708 #45 Mar 05 2021 07:31:42
%S A216708 0,2,2,10,10,18,48,56,86,124,298,336,540,722,1070,2122,2614,3810,5316,
%T A216708 7496,9986,18940,22558,33336,44568,63074,82034,114754,187642,234690,
%U A216708 328536,441872,602006,794020,1072546,1389408,2207532,2706266,3752462,4900474,6681022,8574906
%N A216708 Number of compositions (ordered partitions) of n into 2 or more distinct nonnegative parts.
%C A216708 If permutations are considered equivalent then a(n)=A087135(n)=2*A000009(n) for n>0.
%C A216708 All terms are even. - _Alois P. Heinz_, Aug 18 2018
%H A216708 Alois P. Heinz, <a href="/A216708/b216708.txt">Table of n, a(n) for n = 0..5000</a>
%H A216708 César Eliud Lozada, <a href="/A216708/a216708.pdf">Illustration for terms up to n=9</a>
%F A216708 From _Joerg Arndt_, Sep 17 2012: (Start)
%F A216708 G.f. sum(k>=0, (k+1)!*x^((k^2+k)/2) / prod(j=1..k+1, 1-x^j)) - 1/(1-x);
%F A216708 explanation: the g.f. for partitions into at least two positive parts (A111133) is
%F A216708 sum(k>=0, x^((k^2+k)/2) / prod(j=1..k, 1-x^j)) - 1/(1-x)
%F A216708 (i.e., the g.f. of A000009 minus the g.f. 1/(1-x) for the constant sequence a(n)=1 that counts the single partition n = [n]);
%F A216708 the factor (k+1)! in the g.f. of this function provides for the permutations of the parts, including a zero.
%F A216708 (End)
%e A216708 a(2)=2 because 2 = 0+2 = 2+0 (2 ways)
%e A216708 a(3)=10 because 3 = 0+3 = 1+2 = 2+1 = 3+0 = 0+1+2 = 0+2+1 = 1+0+2 = 1+2+0 = 2+0+1 = 2+1+0 (10 ways)
%p A216708 b:= proc(n, i, p) option remember; (m-> `if`(m<n, 0, `if`(n=0,
%p A216708       `if`(p=0, 0, `if`(p=1, 2, p!*(p+2))), b(n, i-1, p)+
%p A216708         b(n-i, min(n-i, i-1), p+1))))(i*(i+1)/2)
%p A216708     end:
%p A216708 a:= n-> b(n$2, 0):
%p A216708 seq(a(n), n=0..42);  # _Alois P. Heinz_, Aug 18 2018
%t A216708 b[n_, i_, p_] := b[n, i, p] = With[{m = i(i+1)/2}, If[m < n, 0, If[n == 0,
%t A216708      If[p == 0, 0, If[p == 1, 2, p! (p+2)]], b[n, i-1, p] +
%t A216708      b[n-i, Min[n-i, i-1], p+1]]]];
%t A216708 a[n_] := b[n, n, 0];
%t A216708 a /@ Range[0, 42] (* _Jean-François Alcover_, Mar 05 2021, after _Alois P. Heinz_ *)
%o A216708 (PARI)
%o A216708 N=66;  x='x+O('x^N);
%o A216708 gf=sum(k=0,N, (k+1)!*x^((k^2+k)/2) / prod(j=1,k+1, 1-x^j)) - 1/(1-x);
%o A216708 v=Vec(gf);
%o A216708 vector(#v+1,n,if(n==1,0,v[n-1]))
%o A216708 /* _Joerg Arndt_, Sep 17 2012 */
%Y A216708 Cf. A216695, A087135, A000009, A032020.
%K A216708 nonn
%O A216708 0,2
%A A216708 _César Eliud Lozada_, Sep 16 2012
%E A216708 More terms, _Joerg Arndt_, Sep 17 2012