A116665 - OEIS

%I #14 Mar 07 2016 05:44:01

%S 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 210,248,293,345,405,474,555,645,751,872,1009,1166,1346,1549,1781,

%U 2044,2341,2678,3060,3488,3973,4520,5132,5822,6597,7464,8436,9525,10740

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

%C a(n) = Sum(k*A116664(n,k), k>=0).

%H Alois P. Heinz, <a href="/A116665/b116665.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: x(1-x+x^2)/[(1-x^4)product(1-x^(2j-1), j=1..infinity)].

%F a(n) ~ 3^(1/4) * exp(sqrt(n/3)*Pi) / (8*Pi*n^(1/4)). - _Vaclav Kotesovec_, Mar 07 2016

%e 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 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 # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],

%p       `if`(i<1, 0, add((p-> p+`if`(j=1, [0, p[1]], 0))

%p        (b(n-i*j, i-2)), j=0..n/i)))

%p     end:

%p a:= n-> b(n, n-irem(n+1, 2))[2]:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 16 2014

%t 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 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 Cf. A116664.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Feb 22 2006