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

0, 2, 2, 10, 10, 18, 48, 56, 86, 124, 298, 336, 540, 722, 1070, 2122, 2614, 3810, 5316, 7496, 9986, 18940, 22558, 33336, 44568, 63074, 82034, 114754, 187642, 234690, 328536, 441872, 602006, 794020, 1072546, 1389408, 2207532, 2706266, 3752462, 4900474, 6681022, 8574906

Offset: 0

Author

César Eliud Lozada, Sep 16 2012

Keywords

• nonn

Comments

If permutations are considered equivalent then a(n)=A087135(n)=2*A000009(n) for n>0.

All terms are even. - Alois P. Heinz, Aug 18 2018

Examples

			a(2)=2 because 2 = 0+2 = 2+0 (2 ways)
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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000
• César Eliud Lozada, Illustration for terms up to n=9

Crossrefs

Cf. A216695, A087135, A000009, A032020.

Programs

Maple

b:= proc(n, i, p) option remember; (m-> `if`(m b(n$2, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 18 2018

Mathematica

b[n_, i_, p_] := b[n, i, p] = With[{m = i(i+1)/2}, If[m < n, 0, If[n == 0,
     If[p == 0, 0, If[p == 1, 2, p! (p+2)]], b[n, i-1, p] +
     b[n-i, Min[n-i, i-1], p+1]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 42] (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

PARI

N=66;  x='x+O('x^N);
gf=sum(k=0,N, (k+1)!*x^((k^2+k)/2) / prod(j=1,k+1, 1-x^j)) - 1/(1-x);
v=Vec(gf);
vector(#v+1,n,if(n==1,0,v[n-1]))
/* Joerg Arndt, Sep 17 2012 */

Formula

From Joerg Arndt, Sep 17 2012: (Start)

G.f. sum(k>=0, (k+1)!*x^((k^2+k)/2) / prod(j=1..k+1, 1-x^j)) - 1/(1-x);

explanation: the g.f. for partitions into at least two positive parts (A111133) is

sum(k>=0, x^((k^2+k)/2) / prod(j=1..k, 1-x^j)) - 1/(1-x)

(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]);

the factor (k+1)! in the g.f. of this function provides for the permutations of the parts, including a zero.

(End)

Extensions

More terms, Joerg Arndt, Sep 17 2012