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

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

C. G. Bower, Transforms (2)

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

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 *)

Prog

(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 */

Crossrefs

Cf. A032020, A000700.

Sequence in context: A331982 A106380 A076198 * A306703 A295686 A246996

Adjacent sequences: A032018 A032019 A032020 * A032022 A032023 A032024

Keyword

nonn

Author

Christian G. Bower

Extensions

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

Status

approved