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

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, 210, 248, 293, 345, 405, 474, 555, 645, 751, 872, 1009, 1166, 1346, 1549, 1781, 2044, 2341, 2678, 3060, 3488, 3973, 4520, 5132, 5822, 6597, 7464, 8436, 9525, 10740

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

Cf. A116664.

Sequence in context: A082538 A035939 A365827 * A122135 A339572 A027194

Adjacent sequences: A116662 A116663 A116664 * A116666 A116667 A116668

Keyword

nonn

Author

Emeric Deutsch, Feb 22 2006

Status

approved