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
(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 a<b and (a^b)&1) # Chai Wah Wu, Dec 14 2024
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Emeric Deutsch, Apr 05 2006
STATUS
approved