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A058647 McKay-Thompson series of class 36D for the Monster simple group. 8
1, 0, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 47, 60, 76, 96, 120, 150, 185, 228, 280, 342, 416, 504, 608, 732, 878, 1050, 1252, 1488, 1765, 2088, 2464, 2901, 3408, 3996, 4676, 5460, 6364, 7404, 8600, 9972, 11545, 13344, 15400, 17748, 20424, 23472, 26938, 30876 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

G. C. Greubel, Table of n, a(n) for n = -1..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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1) * f(q^3) * phi(q^3) / (phi(-q^2) * psi(q^9)) in powers of q where f(), phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 13 2011

Expansion of eta(q^4) * eta(q^6)^8 * eta(q^9) / (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3 * eta(q^18)^2) in powers of q. - Michael Somos, Feb 13 2011

Euler transform of period 36 sequence [ 0, 2, 3, 1, 0, -3, 0, 1, 2, 2, 0, -1, 0, 2, 3, 1, 0, -2, 0, 1, 3, 2, 0, -1, 0, 2, 2, 1, 0, -3, 0, 1, 3, 2, 0, 0, ...]. - Michael Somos, Feb 13 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 13 2011

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Dec 03 2015

EXAMPLE

T36D = 1/q + 2*q + 3*q^2 + 4*q^3 + 6*q^4 + 9*q^5 + 12*q^6 + 16*q^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 2 q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 3, 0, q^3] / (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

nmax = 60; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(3*k))^2 * (1+x^(3*k))^5 / ((1-x^(2*k)) * (1+x^(6*k))^3 * (1-x^(9*k)) * (1+x^(9*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 03 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^8 * eta(x^9 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3 * eta(x^18 + A)^2), n))} /* Michael Somos, Feb 13 2011 */

CROSSREFS

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

Sequence in context: A073576 A186115 A187020 * A305630 A271147 A069907

Adjacent sequences:  A058644 A058645 A058646 * A058648 A058649 A058650

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

STATUS

approved

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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)