OFFSET
1,2
COMMENTS
Define f(q) = (eta(q^2)*eta(q^3))^7/(eta(q)*eta(q^6))^5, g(q) = Sum_{n>=1} a(n)/n^3 * q^n and t(q) = (eta(q)*eta(q^6)/(eta(q^2)*eta(q^3))^12.
And define the sequence {b(n)} = {0, 6, 351/4, 62531/36, ...} as the solutions of the recursion (n+1)^3*b(n+1) = (34*n^3 + 51*n^2 + 27*n +5)*b(n) - n^3*b(n-1), n >= 1 with b(0) = 0, b(1) = 6.
The following equation holds: 6*f(q)*g(q) = Sum_{n>=0} b(n)*t(q)^n.
REFERENCES
D. Zagier, "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Wikipedia, Apéry's theorem.
EXAMPLE
6*f(q)*g(q)
= 6*(1 + 5*q + 13*q^2 + 23*q^3 + 29*q^4 + 30*q^5 + 31*q^6 + 40*q^7 + ... )
*(q - 19/8*q^2 + 91/27*q^3 - 179/64*q^4 + 126/125*q^5 - 1/216*q^6 + 344/343q^7 - ... )
= 6*q + 63/4*q^2 + 971/36*q^3 + 10679/288*q^4 + 1126103/36000*q^5 + 105401/2400*q^6 + 536870027/12348000*q^7 + ...
= 6 * (q - 12*q^2 + 66*q^3 - 220*q^4 + 495*q^5 - 804*q^6 + 1068*q^7 - ... )
+ 351/4 * (q^2 - 24*q^3 + 276*q^4 - 2024*q^5 + 10626*q^6 - 42528*q^7 + ... )
+ 62531/36 * (q^3 - 36*q^4 + 630*q^5 - 7140*q^6 + 58905*q^7 - ... )
+ 11424695/288 * (q^4 - 48*q^5 + 1128*q^6 - 17296*q^7 + ... )
+ 35441662103/36000 * (q^5 - 60*q^6 + 1770*q^7 - ... )
+ ...
MATHEMATICA
a[n_Integer] := Module[{b, ary}, b = Join[{0}, Table[DivisorSigma[3, i], {i, 1, n}]]; ary = b; Do[If[Mod[i, 2] == 0, ary[[i + 1]] -= 28*b[[i/2 + 1]]]; If[Mod[i, 3] == 0, ary[[i + 1]] += 63*b[[i/3 + 1]]]; If[Mod[i, 6] == 0, ary[[i + 1]] -= 36*b[[i/6 + 1]]]; , {i, 1, n}]; Rest[ary]]; a[38] (* Robert P. P. McKone, Aug 23 2023 *)
PROG
(Ruby)
def A001158(n)
s = 0
(1..n).each{|i| s += i * i * i if n % i == 0}
s
end
def A288485(n)
a = [0] + (1..n).map{|i| A001158(i)}
ary = a.clone
(1..n).each{|i|
ary[i] -= 28 * a[i / 2] if i % 2 == 0
ary[i] += 63 * a[i / 3] if i % 3 == 0
ary[i] -= 36 * a[i / 6] if i % 6 == 0
}
ary[1..-1]
end
p A288485(100)
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jun 09 2017
STATUS
approved