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
Keywords
Examples
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].
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.
Programs
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Magma
m:=80; R
:=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 -
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);
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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 *) -
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 -
Python
from sympy import divisors 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 aChai Wah Wu, Dec 14 2024
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