A117956 - OEIS

%I #25 Sep 08 2022 08:45:24

%S 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 60,88,60,96,76,106,76,122,93,126,94,140,111,158,106,163,134,175,127,

%U 196,150,198,149,212,170,240,164,238,200,250,180,284,214,277,216,292,238

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

%H Alois P. Heinz, <a href="/A117956/b117956.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 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 G.f.: Sum_{i>=1}(Sum{j>=1}(x^(2*i+2*j-1)/((1-x^(2*i-1))*(1-x^(2*j)))).

%F 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 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 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 gser:=series(g, x=0, 70): seq(coeff(gser, x^n), n=1..67);

%t 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 (PARI) 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 (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

%Y Cf. A002133, A117955.

%K nonn

%O 1,5

%A _Emeric Deutsch_, Apr 05 2006