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A067553
Sum of products of terms in all partitions of n into odd parts.
14
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
KEYWORD
easy,nonn
AUTHOR
Naohiro Nomoto, Jan 29 2002
EXTENSIONS
Corrected a(0) from 0 to 1, Joerg Arndt, Oct 06 2012
STATUS
approved