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

%I A067553 #21 Dec 15 2015 18:14:13
%S A067553 1,1,1,4,4,9,18,25,40,76,122,178,321,472,734,1303,1874,2852,4782,6984,
%T A067553 10808,17552,25461,38512,61586,90894,135437,213260,312180,463340,
%U A067553 728806,1057468,1562810,2422394,3511962,5215671,7985196,11550542,17022228,25924746,37638033
%N A067553 Sum of products of terms in all partitions of n into odd parts.
%C A067553 a(0) = 1 as the empty product equals 1. [_Joerg Arndt_, Oct 06 2012]
%H A067553 Alois P. Heinz and Vaclav Kotesovec, <a href="/A067553/b067553.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..1000 from Alois P. Heinz)
%F A067553 G.f.: 1/(Product_{k>=0} (1-(2*k+1)*x^(2*k+1)) ). - _Vladeta Jovovic_, May 09 2003
%F A067553 From _Vaclav Kotesovec_, Dec 15 2015: (Start)
%F A067553 a(n) ~ c * 3^(n/3), where
%F A067553 c = 28.8343667894061364904068323836801301428320806272385991... if mod(n,3) = 0
%F A067553 c = 28.4762018725001067057188975211539643762050439184376103... if mod(n,3) = 1
%F A067553 c = 28.3618072960214990676207117911869616961300790076910101... if mod(n,3) = 2.
%F A067553 (End)
%p A067553 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A067553       b(n, i-1) +`if`(i>n or irem(i, 2)=0, 0, i*b(n-i, i))))
%p A067553     end:
%p A067553 a:= n-> b(n$2):
%p A067553 seq(a(n), n=0..50);  # _Alois P. Heinz_, Sep 07 2014
%t A067553 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_ *)
%t A067553 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 *)
%o A067553 (PARI)
%o A067553 N=66; q='q+O('q^N);
%o A067553 gf= 1/ prod(n=1,N, (1-(2*n-1)*q^(2*n-1)) );
%o A067553 Vec(gf)
%o A067553 /* _Joerg Arndt_, Oct 06 2012 */
%o A067553 (Maxima)
%o A067553 g(n):= if n=0 then 1 else if oddp(n)=true  then n else 0;
%o A067553 P(m,n):=if n=m then g(n) else sum(g(k)*P(k,n-k),k,m,n/2)+g(n);
%o A067553 a(n):=P(1,n);
%o A067553 makelist(a(n),n,0,27); /* _Vladimir Kruchinin_, Sep 06 2014 */
%Y A067553 Cf. A006906, A022629, A000009.
%K A067553 easy,nonn
%O A067553 0,4
%A A067553 _Naohiro Nomoto_, Jan 29 2002
%E A067553 Corrected a(0) from 0 to 1, _Joerg Arndt_, Oct 06 2012