A117955 Number of partitions of n into exactly 2 types of odd parts.

0, 0, 0, 1, 1, 2, 3, 5, 4, 7, 8, 10, 11, 13, 12, 19, 18, 20, 22, 25, 24, 30, 31, 36, 33, 39, 38, 45, 45, 48, 51, 57, 54, 60, 56, 69, 67, 72, 72, 79, 78, 84, 84, 90, 87, 97, 97, 112, 99, 107, 112, 117, 115, 126, 118, 131, 134, 137, 136, 152, 143, 149, 149, 163, 152, 174, 164

Offset: 1

Author

Emeric Deutsch, Apr 05 2006

Keywords

• nonn

Examples

			a(8)=5 because we have [7,1],[5,3],[5,1,1,1],[3,3,1,1] and [3,3,1,1].
		

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.

Crossrefs

Cf. A002133.

Programs

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 *)

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 aChai Wah Wu, Dec 14 2024

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