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A226054
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McKay-Thompson series of class 45A for the Monster group with a(0) = 1.
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2
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1, 1, 2, 1, 3, 4, 5, 6, 7, 11, 15, 17, 22, 24, 34, 40, 48, 56, 69, 84, 104, 118, 144, 164, 200, 234, 273, 318, 372, 436, 511, 582, 681, 775, 906, 1036, 1192, 1362, 1562, 1784, 2046, 2315, 2647, 2988, 3409, 3860, 4371, 4936, 5573, 6288, 7104, 7967, 8979, 10052
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OFFSET
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-1,3
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LINKS
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FORMULA
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Expansion of (eta(q^3) * eta(q^15))^2 / (eta(q) * eta(q^5) * eta(q^9) * eta(q^45)) in powers of q.
Euler transform of period 45 sequence [ 1, 1, -1, 1, 2, -1, 1, 1, 0, 2, 1, -1, 1, 1, -2, 1, 1, 0, 1, 2, -1, 1, 1, -1, 2, 1, 0, 1, 1, -2, 1, 1, -1, 1, 2, 0, 1, 1, -1, 2, 1, -1, 1, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u*v + 3) - (u+v) * (u^2 + u*v + v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (45 t)) = f(t) where q = exp(2 Pi i t).
a(n) ~ exp(4*Pi*sqrt(n/5)/3) / (5^(1/4) * sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
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EXAMPLE
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1/q + 1 + 2*q + q^2 + 3*q^3 + 4*q^4 + 5*q^5 + 6*q^6 + 7*q^7 + 11*q^8 + ...
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MATHEMATICA
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nmax=60; CoefficientList[Series[Product[(1-x^(3*k))^2 * (1-x^(15*k))^2 / ((1-x^k) * (1-x^(5*k)) * (1-x^(9*k)) * (1-x^(45*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q^3]*eta[q^15])^2/(eta[q]*eta[q^5]*eta[q^9]*eta[q^45]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 20 2018 *)
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PROG
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(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^15 + A))^2 / (eta(x + A) * eta(x^5 + A) * eta(x^9 + A) * eta(x^45 + A)), 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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