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A116665 Total number of parts that appear exactly once in the partitions of n into odd parts.

%I A116665 #14 Mar 07 2016 05:44:01
%S A116665 0,1,0,1,2,2,3,4,6,7,10,12,16,20,25,31,39,47,58,71,85,103,124,148,176,
%T A116665 210,248,293,345,405,474,555,645,751,872,1009,1166,1346,1549,1781,
%U A116665 2044,2341,2678,3060,3488,3973,4520,5132,5822,6597,7464,8436,9525,10740
%N A116665 Total number of parts that appear exactly once in the partitions of n into odd parts.
%C A116665 a(n) = Sum(k*A116664(n,k), k>=0).
%H A116665 Alois P. Heinz, <a href="/A116665/b116665.txt">Table of n, a(n) for n = 0..1000</a>
%F A116665 G.f.: x(1-x+x^2)/[(1-x^4)product(1-x^(2j-1), j=1..infinity)].
%F A116665 a(n) ~ 3^(1/4) * exp(sqrt(n/3)*Pi) / (8*Pi*n^(1/4)). - _Vaclav Kotesovec_, Mar 07 2016
%e A116665 a(8) = 6 because in the partitions of 8 into odd parts, namely, [(7),(1)], [(5),(3)], [(5),1,1,1], [3,3,1,1], [(3),1,1,1,1,1] and [1,1,1,1,1,1,1,1], we have 6 parts that appear exactly once (shown between parentheses).
%p A116665 f:=x*(1-x+x^2)/(1-x^4)/product(1-x^(2*j-1),j=1..40): fser:=series(f,x=0,61): seq(coeff(fser,x,n),n=0..57);
%p A116665 # second Maple program:
%p A116665 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A116665       `if`(i<1, 0, add((p-> p+`if`(j=1, [0, p[1]], 0))
%p A116665        (b(n-i*j, i-2)), j=0..n/i)))
%p A116665     end:
%p A116665 a:= n-> b(n, n-irem(n+1, 2))[2]:
%p A116665 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 16 2014
%t A116665 b[n_, i_] :=  b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[{p}, p + If[j == 1, {0, p[[1]]}, 0]][b[n-i*j, i-2]], {j, 0, n/i}]]]; a[n_] := b[n, n - Mod[n+1, 2]][[2]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 13 2015, after _Alois P. Heinz_ *)
%t A116665 nmax = 60; CoefficientList[Series[x*(1-x+x^2)/(1-x^4) * Product[1/(1-x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 07 2016 *)
%Y A116665 Cf. A116664.
%K A116665 nonn
%O A116665 0,5
%A A116665 _Emeric Deutsch_, Feb 22 2006