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

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, 60, 88, 60, 96, 76, 106, 76, 122, 93, 126, 94, 140, 111, 158, 106, 163, 134, 175, 127, 196, 150, 198, 149, 212, 170, 240, 164, 238, 200, 250, 180, 284, 214, 277, 216, 292, 238

Offset

1,5

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 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. Int. 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.

Formula

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

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

Example

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].

Maple

g := add(add(x^(2*i+2*j-1)/(1-x^(2*i-1))/(1-x^(2*j)), j=1..70), i=1..70):
gser:=series(g, x=0, 70): seq(coeff(gser, x^n), n=1..67);

Mathematica

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 *)

Prog

(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
(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

Crossrefs

Cf. A002133, A117955.

Sequence in context: A011451 A200089 A330686 * A241638 A360689 A110662

Adjacent sequences: A117953 A117954 A117955 * A117957 A117958 A117959

Keyword

nonn

Author

Emeric Deutsch, Apr 05 2006

Status

approved