cp's OEIS Frontend

A117956 Number of partitions of n into exactly 2 types of parts: one odd and one even.

%I A117956 #33 Jan 24 2025 08:50:33
%S A117956 0,0,1,1,4,3,8,6,13,10,19,13,26,20,32,23,41,31,49,34,58,45,66,47,76,
%T A117956 60,88,60,96,76,106,76,122,93,126,94,140,111,158,106,163,134,175,127,
%U A117956 196,150,198,149,212,170,240,164,238,200,250,180,284,214,277,216,292,238
%N A117956 Number of partitions of n into exactly 2 types of parts: one odd and one even.
%H A117956 Alois P. Heinz, <a href="/A117956/b117956.txt">Table of n, a(n) for n = 1..10000</a>
%H A117956 D. Christopher and T. Nadu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Christopher/chris7.html">Partitions with Fixed Number of Sizes</a>, Journal of Integer Sequences, 15 (2015), #15.11.5.
%H A117956 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. Int. Seq. 14 (2011) # 11.3.6.
%H A117956 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.
%F A117956 G.f.: Sum_{i>=1} Sum{j>=1} x^(2*i+2*j-1)/((1-x^(2*i-1))*(1-x^(2*j))).
%F A117956 Convolution of x(n) and y(n), where x(n) is the number of even divisors of n and y(n) is the number of odd divisors of n. - _Vladeta Jovovic_, Apr 05 2006
%e A117956 a(7) = 8 because we have [6,1], [5,2], [4,3], [4,1,1,1], [3,2,2], [2,2,2,1],[2,2,1,1,1] and [2,1,1,1,1,1].
%p A117956 g := add(add(x^(2*i+2*j-1)/(1-x^(2*i-1))/(1-x^(2*j)), j=1..70), i=1..70):
%p A117956 gser:=series(g, x=0, 70): seq(coeff(gser, x^n), n=1..67);
%t A117956 With[{nmax = 80}, CoefficientList[Series[Sum[Sum[x^(2*k + 2*j - 2)/((1 - x^(2*k - 1))*(1 - x^(2*j))), {j, 1, 2*nmax}], {k, 1, 2*nmax}], {x, 0, nmax}], x]] (* _G. C. Greubel_, Oct 06 2018 *)
%o A117956 (PARI) my(x='x+O('x^80)); concat([0,0], Vec(sum(k=1,100, sum(j=1,100, x^(2*k + 2*j - 2)/((1 - x^(2*k - 1))*(1 - x^(2*j))))))) \\ _G. C. Greubel_, Oct 06 2018
%o A117956 (Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!((&+[(&+[x^(2*k + 2*j - 2)/((1 - x^(2*k - 1))*(1 - x^(2*j))): j in [1..100]]): k in [1..100]]))); // _G. C. Greubel_, Oct 06 2018
%o A117956 (Python)
%o A117956 from sympy import divisors
%o A117956 def A117956(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 and (a^b)&1) # _Chai Wah Wu_, Dec 14 2024
%Y A117956 Cf. A002133, A117955.
%K A117956 nonn
%O A117956 1,5
%A A117956 _Emeric Deutsch_, Apr 05 2006