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A163129
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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.
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5
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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