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A153084
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McKay-Thompson series of class 15C for the Monster group with a(0) = 3.
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3
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1, 3, 9, 19, 42, 78, 146, 249, 429, 695, 1125, 1749, 2713, 4086, 6123, 8986, 13122, 18852, 26934, 38001, 53328, 74068, 102336, 140208, 191153, 258741, 348606, 466806, 622383, 825342, 1090087, 1432923, 1876542, 2447029, 3179859, 4116282, 5311204, 6829008
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OFFSET
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-1,2
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COMMENTS
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Equals the triangular series convolved with the aerated variant of A153084; i.e., (1, 3, 6, 10, ...) * (1, 0, 3, 0, 9, ...). Note that (1, 3, 6, 10, ...) convolved with (1, 0, 3, 0, 6, ...) = A038163: (1, 3, 9, 19, 39, 69, ...). - Gary W. Adamson, Aug 12 2016
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LINKS
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FORMULA
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Expansion of (eta(q^3) * eta(q^5) / (eta(q) * eta(q^15)))^3 in powers of q.
Euler transform of period 15 sequence [ 3, 3, 0, 3, 0, 0, 3, 3, 0, 0, 3, 0, 3, 3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v) * (u^2 + 5*u*v + v^2) - u*v * (u*v - 1).
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = f(t) where q = exp(2 Pi i t).
G.f. x^(-1) * (Prod_{k>0} ( (1 - x^(3*k)) * (1 - x^(5*k)) / ((1 - x^k) * (1 - x^(15*k)))))^3. [corrected by Vaclav Kotesovec, Jun 26 2018]
a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 26 2018
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EXAMPLE
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T15C = 1/q + 3 + 9*q + 19*q^2 + 42*q^3 + 78*q^4 + 146*q^5 + 249*q^6 + 429*q^7 + ...
42 = (1, 3, 6, 10, 15) dot (9, 0, 3, 0, 1) = (9 + 0 + 18 + 0 + 15). - Gary W. Adamson, Aug 12 2016
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MATHEMATICA
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nmax=60; CoefficientList[Series[Product[(((1-x^(3*k)) * (1-x^(5*k)) / (1-x^(15*k)) / (1-x^k)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015, typo corrected by Vaclav Kotesovec, Jun 26 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^5 + A) / (eta(x + A) * eta(x^15 + A)))^3, 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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