A288485 Expansion of (E_4(q) - 28*E_4(q^2) + 63*E_4(q^3) - 36*E(q^6)) / 240.

1, -19, 91, -179, 126, -1, 344, -1459, 2521, -2394, 1332, -737, 2198, -6536, 11466, -11699, 4914, 485, 6860, -22554, 31304, -25308, 12168, -6625, 15751, -41762, 68131, -61576, 24390, -126, 29792, -93619, 121212, -93366, 43344, -15803, 50654, -130340

Offset: 1

Author

Seiichi Manyama, Jun 09 2017

Keywords

• sign

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.

Examples

			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 - ... )
  + ...
		

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.

Crossrefs

{b(n)} = {A059415(n)/A059416(n)} = {0, 6, 351/4, 62531/36, ...}.

Cf. A001158 (sigma_3(n)), A004009 (E_4), A006353 (f(q)), A226235 (t(q)).

Programs

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 *)

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)