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
Links
- Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)
Programs
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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 *) -
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 */
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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 */
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)
Extensions
Corrected a(0) from 0 to 1, Joerg Arndt, Oct 06 2012
Comments