A163129 G.f.: A(q) = exp( Sum_{n>=1} sigma(n) * 3*A038500(n) * q^n/n ), where A038500(n) = highest power of 3 dividing n.
1, 3, 9, 30, 75, 180, 441, 969, 2070, 4431, 8964, 17775, 35094, 66975, 125865, 235053, 429096, 773766, 1386027, 2442372, 4260645, 7384578, 12640320, 21453975, 36192519, 60454713, 100250100, 165311094, 270391857, 439479198, 710631279
Offset: 0
Keywords
Examples
G.f.: A(q) = 1 + 3*q + 9*q^2 + 30*q^3 + 75*q^4 + 180*q^5 + 441*q^6 + ... log(A(q)) = 3*q + 9*q^2/2 + 36*q^3/3 + 21*q^4/4 + 18*q^5/5 + 108*q^6/6 + ... Define TRISECTIONS: T_0(q) = 1 + 30*q^3 + 441*q^6 + 4431*q^9 + 35094*q^12 + ... T_1(q) = 3*q + 75*q^4 + 969*q^7 + 8964*q^10 + 66975*q^13 + ... T_2(q) = 9*q^2 + 180*q^5 + 2070*q^8 + 17775*q^11 + 125865*q^14 + ... then: 3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091: T9B(q) = 1/q + 5*q^2 - 7*q^5 + 3*q^8 + 15*q^11 - 32*q^14 + 9*q^17 + 58*q^20 + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Crossrefs
Programs
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Mathematica
nmax = 100; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*3^(IntegerExponent[k, 3] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], q] (* G. C. Greubel, Jul 03 2018, edited by Vaclav Kotesovec, Oct 20 2020 *) -
PARI
{a(n)=local(L=sum(m=1, n, 3*sigma(m)*3^valuation(m, 3)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}
Formula
Define trisections by: A(q) = T_0(q) + T_1(q) + T_2(q), then:
3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091,
which is the McKay-Thompson series of class 9B for Monster.
G.f.: 1/Product_{n>=0} R(q^(3^n))^(3^n) where R(q) = E(q)^3/E(q^3) and E(q) = Product_{k>=1} (1 - q^k). - Joerg Arndt, Aug 03 2011