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A058601 McKay-Thompson series of class 27b for the Monster group. 5
1, 2, 5, 6, 12, 16, 27, 34, 51, 70, 101, 134, 182, 240, 322, 416, 544, 696, 902, 1144, 1462, 1832, 2317, 2882, 3608, 4454, 5524, 6786, 8352, 10200, 12463, 15136, 18384, 22210, 26826, 32250, 38768, 46408, 55531, 66186, 78859, 93638, 111123, 131462, 155428, 183280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Expansion of q^(1/3) * eta(q^3)^4 / (eta(q) * eta(q^9))^2 in powers of q. - Michael Somos, Jun 30 2011

Expansion of q^(1/3) * ( (c(q) * b(q^3)) / (b(q) * c(q^3)) )^(1/2) in powers of q where b(), c() are cubic AGM functions. - Michael Somos, Jun 30 2011

Euler transform of period 9 sequence [ 2, 2, -2, 2, 2, -2, 2, 2, 0, ...]. - Michael Somos, Jun 30 2011

Given g.f. A(x) then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v) * (u - v^2) + 4*u*v. - Michael Somos, Jun 30 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (81 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 30 2011

G.f.: Product_{k>0} (1 - x^(3*k))^4 / ((1 - x^k) * (1 - x^(9*k)))^2.

a(n) = A058096(3*n - 1) = A152954(3*n - 1).

Convolution inverse of A192329.

Convolution square of A112194. Convolution cube is A131985. - Michael Somos, Aug 28 2015

a(n) ~ exp(4*Pi*sqrt(n/3)/3) / (sqrt(2) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

EXAMPLE

G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 12*x^4 + 16*x^5 + 27*x^6 + 34*x^7 + ...

T27b = 1/q + 2*q^2 + 5*q^5 + 6*q^8 + 12*q^11 + 16*q^14 + 27*q^17 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^9]))^2, {x, 0, n}]; (* Michael Somos, Aug 28 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^4 / (eta(x + A) * eta(x^9 + A))^2, n))}; /* Michael Somos, Jun 30 2011 */

CROSSREFS

Cf. A000521, A007240, A014708, A007241, A007267, A045478, A192329.

Cf. A058096, A112194, A131985, A152954.

Sequence in context: A023143 A085206 A238481 * A108365 A064765 A257805

Adjacent sequences:  A058598 A058599 A058600 * A058602 A058603 A058604

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

STATUS

approved

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Last modified January 20 17:05 EST 2019. Contains 319335 sequences. (Running on oeis4.)