A032021 Number of compositions (ordered partitions) of n into distinct odd parts.

1, 1, 0, 1, 2, 1, 2, 1, 4, 7, 4, 7, 6, 13, 6, 19, 32, 25, 32, 31, 58, 43, 82, 49, 132, 181, 156, 193, 230, 325, 278, 457, 376, 715, 448, 967, 1290, 1345, 1386, 1723, 2276, 2341, 3116, 2959, 4750, 3823, 6358, 4681, 9480, 10945, 11832, 12169, 16442, 18793, 21002, 25537, 27820, 37687

Offset: 0

Author

Christian G. Bower

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A032020, A000700.

Programs

Maple

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
       ->x+y, b(n, i-2), [0, `if`(i>n, [], b(n-i, i-2))[]], 0)))
    end:
a:= proc(n) local l; l:= b(n, n-1+irem(n, 2));
      a(n):= add(l[i]*(i-1)!, i=1..nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 09 2012

Mathematica

b[n_, i_] := If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-2], Join[{0}, If[i>n, {}, b[n-i, i-2]]]}]]]; a[n_] := Module[{l}, l = b[n, n-1+Mod[n, 2]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

PARI

N=66;  q='q+O('q^N);
gf=sum(k=0,N, k! * q^(k^2) / prod(j=1,k, 1-q^(2*j) ) );
Vec(gf)
/* Joerg Arndt, Sep 17 2012 */

Formula

"AGK" (ordered, elements, unlabeled) transform of 1, 0, 1, 0...(odds)

G.f.: sum(k>=0, k! * x^(k^2) / prod(j=1..k, 1-x^(2*j) ) ). - Vladeta Jovovic, Aug 05 2004

Extensions

Prepended a(0)=1, Joerg Arndt, Oct 20 2012