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A058092
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McKay-Thompson series of class 9a for the Monster group.
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7
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1, 14, 65, 156, 456, 1066, 2250, 4720, 9426, 17590, 32801, 58904, 102650, 176646, 298066, 491792, 803923, 1293450, 2051156, 3221716, 5004028, 7682744, 11703580, 17663312, 26423351, 39248618, 57866503, 84685920, 123188502, 178054416, 255782770, 365467216
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OFFSET
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0,2
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COMMENTS
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In volume 2 of Raamunjuan's Notebooks is an obscure equation involving t(1-t) on the left and GG' on the right and they both are equal to the g.f. of 1/3 of this sequence. Here t^(1/3) = c(x)/a(x), (1-t)^(1/3) = b(x)/a(x) since a(x)^3 = b(x)^3 + c(x)^3. N.B. The left side was (t(1-t))^(1/3) but the exponent should be (-1/3) instead which is why the equation was so obscure. - Michael Somos, Mar 13 2019
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 179.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 392.
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LINKS
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FORMULA
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Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/3) in powers of x where b(), c() are cubic AGM theta functions, Michael Somos, Jun 16 2012
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EXAMPLE
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G.f. = 1 + 14*x + 65*x^2 + 156*x^3 + 456*x^4 + 1066*x^5 + 2250*x^6 + 4720*x^7 + ...
T9a = 1/q + 14*q^2 + 65*q^5 + 156*q^8 + 456*q^11 + 1066*q^14 + 2250*q^17 + ...
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MATHEMATICA
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a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, A = (QPochhammer[x^3 + A] / QPochhammer[x + A])^12; SeriesCoefficient[((1 + 27*x*A)^2/A)^(1/3), n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 06 2015, adapted from Michael Somos's PARI script *)
CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3)^2 / (QPochhammer[x, x]^2*QPochhammer[x^3, x^3]^4), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 07 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( ((1 + 27 * x * A)^2 / A)^(1/3), n))}; /* Michael Somos, Jun 16 2012 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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