A104274 Number of partitions of n in which odd squares occur with 2 types c,c* and with multiplicity 1. The even squares and parts that are twice the squares they occur with multiplicity 1.
1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 10, 10, 10, 10, 10, 10, 12, 14, 16, 18, 18, 18, 18, 18, 18, 22, 26, 28, 30, 30, 30, 30, 30, 30, 34, 38, 40, 42, 42, 42, 44, 48, 50, 54, 58, 60, 62, 62, 62, 66, 74, 78, 82, 86, 88, 90, 90, 90
Offset: 0
Examples
E.g. a(10)=6 because we can write it as 91,91*,9*1,9*1*,82,811*.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Programs
-
Maple
series(product((1+x^((2*k-1)^2))/(1-x^(2*k-1)^2)),k=1..100),x=0,100);
-
Mathematica
nmax = 100; CoefficientList[Series[Product[(1 + x^((2*k-1)^2)) / (1 - x^((2*k-1)^2)), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 19 2017 *)
Formula
G.f.: product_{k>0}((1+x^(2k-1)^2)/(1-x^(2k-1)^2)).
a(n) ~ exp(3 * 2^(-8/3) * Pi^(1/3) * ((4-sqrt(2)) * Zeta(3/2))^(2/3) * n^(1/3)) * ((4-sqrt(2)) * Zeta(3/2))^(1/3) / (2^(7/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 19 2017
Comments