A067553 Sum of products of terms in all partitions of n into odd parts.

1, 1, 1, 4, 4, 9, 18, 25, 40, 76, 122, 178, 321, 472, 734, 1303, 1874, 2852, 4782, 6984, 10808, 17552, 25461, 38512, 61586, 90894, 135437, 213260, 312180, 463340, 728806, 1057468, 1562810, 2422394, 3511962, 5215671, 7985196, 11550542, 17022228, 25924746, 37638033

Offset

0,4

Comments

a(0) = 1 as the empty product equals 1. [Joerg Arndt, Oct 06 2012]

Links

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Formula

G.f.: 1/(Product_{k>=0} (1-(2*k+1)*x^(2*k+1)) ). - Vladeta Jovovic, May 09 2003

From Vaclav Kotesovec, Dec 15 2015: (Start)

a(n) ~ c * 3^(n/3), where

c = 28.8343667894061364904068323836801301428320806272385991... if mod(n,3) = 0

c = 28.4762018725001067057188975211539643762050439184376103... if mod(n,3) = 1

c = 28.3618072960214990676207117911869616961300790076910101... if mod(n,3) = 2.

(End)

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n or irem(i, 2)=0, 0, i*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 07 2014

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n || Mod[i, 2] == 0, 0, i*b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 02 2015, after Alois P. Heinz *)
nmax = 40; CoefficientList[Series[Product[1/(1-(2*k-1)*x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)

Prog

(PARI)
N=66; q='q+O('q^N);
gf= 1/ prod(n=1, N, (1-(2*n-1)*q^(2*n-1)) );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */
(Maxima)
g(n):= if n=0 then 1 else if oddp(n)=true  then n else 0;
P(m, n):=if n=m then g(n) else sum(g(k)*P(k, n-k), k, m, n/2)+g(n);
a(n):=P(1, n);
makelist(a(n), n, 0, 27); /* Vladimir Kruchinin, Sep 06 2014 */

Crossrefs

Cf. A006906, A022629, A000009.

Sequence in context: A319646 A214826 A135065 * A112683 A192030 A117879

Adjacent sequences: A067550 A067551 A067552 * A067554 A067555 A067556

Keyword

easy,nonn

Author

Naohiro Nomoto, Jan 29 2002

Extensions

Corrected a(0) from 0 to 1, Joerg Arndt, Oct 06 2012

Status

approved