OFFSET
1,6
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, J. Integer Seq. 15 (2015), #15.11.5.
N. Benyahia Tani and Sadek Bouroubi, Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes, J. Integer Seq. 14 (2011), #11.3.6.
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire 03 (2015), 18-27.
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Elemente der Mathematik 72(2) (2017), 66-74.
FORMULA
G.f.: sum(sum(x^(2i+2j-2)/[(1-x^(2i-1))(1-x^(2j-1))], j=1..i-1), i=1..infinity).
G.f. for number of partitions of n into exactly m types of odd parts is obtained if we substitute x(i) with -Sum_{k>0}(x^(2*n-1)/(x^(2*n-1)-1))^i in the cycle index Z(S(m); x(1),x(2),..,x(m)) of the symmetric group S(m) of degree m. - Vladeta Jovovic, Sep 20 2007
EXAMPLE
a(8)=5 because we have [7,1],[5,3],[5,1,1,1],[3,3,1,1] and [3,3,1,1].
MAPLE
g:=sum(sum(x^(2*i+2*j-2)/(1-x^(2*i-1))/(1-x^(2*j-1)), j=1..i-1), i=1..40): gser:=series(g, x=0, 75): seq(coeff(gser, x^n), n=1..72);
MATHEMATICA
With[{nmax = 60}, CoefficientList[Series[Sum[Sum[x^(2*k+2*j-2)/((1-x^(2*k -1))*(1-x^(2*j-1))), {j, 1, k-1}], {k, 1, 3*nmax}], {x, 0, nmax}], x]] (* G. C. Greubel, Oct 05 2018 *)
PROG
(Python)
from sympy import divisors
def A117955(n): return sum(1 for ax in range(1, n-1) for a in divisors(ax, generator=True) for b in divisors(n-ax, generator=True) if a<b-1 and a&b&1) # Chai Wah Wu, Dec 14 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 05 2006
STATUS
approved