OFFSET
0,2
COMMENTS
All terms are even. - Alois P. Heinz, Aug 18 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000
César Eliud Lozada, Illustration for terms up to n=9
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)
EXAMPLE
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)
MAPLE
b:= proc(n, i, p) option remember; (m-> `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))))(i*(i+1)/2)
end:
a:= n-> 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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
César Eliud Lozada, Sep 16 2012
EXTENSIONS
More terms, Joerg Arndt, Sep 17 2012
STATUS
approved