This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A117955 #37 Dec 15 2024 04:40:01
%S A117955 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,
%T A117955 38,45,45,48,51,57,54,60,56,69,67,72,72,79,78,84,84,90,87,97,97,112,
%U A117955 99,107,112,117,115,126,118,131,134,137,136,152,143,149,149,163,152,174,164
%N A117955 Number of partitions of n into exactly 2 types of odd parts.
%H A117955 Alois P. Heinz, <a href="/A117955/b117955.txt">Table of n, a(n) for n = 1..10000</a>
%H A117955 D. Christopher and T. Nadu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Christopher/chris7.html">Partitions with Fixed Number of Sizes</a>, J. Integer Seq. 15 (2015), #15.11.5.
%H A117955 N. Benyahia Tani and Sadek Bouroubi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Tani/tani7.html">Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes</a>, J. Integer Seq. 14 (2011), #11.3.6.
%H A117955 N. Benyahia Tani, S. Bouroubi, and O. Kihel, <a href="https://liforce.usthb.dz/sites/default/files/2020-11/article3.pdf">An effective approach for integer partitions using exactly two distinct sizes of parts</a>, Bulletin du Laboratoire 03 (2015), 18-27.
%H A117955 N. Benyahia Tani, S. Bouroubi, and O. Kihel, <a href="http://doi.org/10.4171/EM/326">An effective approach for integer partitions using exactly two distinct sizes of parts</a>, Elemente der Mathematik 72(2) (2017), 66-74.
%F A117955 G.f.: sum(sum(x^(2i+2j-2)/[(1-x^(2i-1))(1-x^(2j-1))], j=1..i-1), i=1..infinity).
%F A117955 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
%e A117955 a(8)=5 because we have [7,1],[5,3],[5,1,1,1],[3,3,1,1] and [3,3,1,1].
%p A117955 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);
%t A117955 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 *)
%o A117955 (Python)
%o A117955 from sympy import divisors
%o A117955 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
%Y A117955 Cf. A002133.
%K A117955 nonn
%O A117955 1,6
%A A117955 _Emeric Deutsch_, Apr 05 2006