A058092 McKay-Thompson series of class 9a for the Monster group.

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

Offset: 0

Author

N. J. A. Sloane, Nov 27 2000

Keywords

• nonn
• easy

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

Examples

			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 + ...
		

References

• 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.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
• D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
• G. Manco, How to calculate moduli alpha_3n of the Ramanujan's q_3 theory, Mathematics StackExchange, Jan 2017.
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A030197, A051273.

Programs

Mathematica

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 *)

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 */

Formula

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

Convolution cube is A030197.

a(n) ~ exp(4*Pi*sqrt(n)/3) / (sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Nov 07 2015