OFFSET
0,5
COMMENTS
a(n) = Sum(k*A116664(n,k), k>=0).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: x(1-x+x^2)/[(1-x^4)product(1-x^(2j-1), j=1..infinity)].
a(n) ~ 3^(1/4) * exp(sqrt(n/3)*Pi) / (8*Pi*n^(1/4)). - Vaclav Kotesovec, Mar 07 2016
EXAMPLE
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).
MAPLE
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);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, 0, add((p-> p+`if`(j=1, [0, p[1]], 0))
(b(n-i*j, i-2)), j=0..n/i)))
end:
a:= n-> b(n, n-irem(n+1, 2))[2]:
seq(a(n), n=0..60); # Alois P. Heinz, Mar 16 2014
MATHEMATICA
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 *)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 22 2006
STATUS
approved