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A058639 McKay-Thompson series of class 34a for the Monster group. 2
1, 1, 3, 4, 6, 7, 13, 16, 22, 29, 40, 50, 69, 83, 110, 136, 174, 214, 272, 332, 413, 502, 618, 748, 915, 1095, 1329, 1590, 1910, 2272, 2718, 3216, 3823, 4508, 5332, 6262, 7378, 8630, 10119, 11802, 13784, 16023, 18650, 21612, 25070, 28972, 33502, 38610 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Expansion of (psi(q^2) * phi(q^17) - q^4 * phi(q) * psi(q^34)) / (f(-q) * f(-q^17)) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Dec 11 2008

Expansion of (F(q) - q^4 / F(q)) / (chi(-q) * chi(-q^17))^2 in powers of q where F(q) = G(q^17) / G(q), G(q) = chi(q) * chi(-q^2) and chi() is a Ramanujan theta function. - Michael Somos, Dec 11 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (68 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 11 2008

a(n) ~ exp(2*Pi*sqrt(2*n/17)) / (2^(3/4) * 17^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018

EXAMPLE

T34a = 1/q + q + 3*q^3 + 4*q^5 + 6*q^7 + 7*q^9 + 13*q^11 + 16*q^13 + ...

MATHEMATICA

QP = QPochhammer; s = QP[q^4]^2*(QP[q^34]^5/(QP[q]*QP[q^2]*QP[q^17]^3* QP[q^68]^2))-q^4*QP[q^2]^5*(QP[q^68]^2/(QP[q]^3*QP[q^4]^2*QP[q^17]* QP[q^34]))+O[q]^50; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 13 2015, adapted from PARI *)

nmax = 60; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k))^2 * (1+x^(17*k)) * (1+x^(34*k-17))^2, {k, 1, nmax}] - x^4*Product[(1+x^k) * (1+x^(2*k-1))^2 * (1+x^(17*k)) * (1+x^(34*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 01 2017 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^17 + A)^3 * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^68 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^17 + A) * eta(x^34 + A)), n))} /* Michael Somos, Dec 15 2008 */

CROSSREFS

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

Convolution square is A152944.

Sequence in context: A276783 A288731 A093707 * A253888 A289811 A161001

Adjacent sequences:  A058636 A058637 A058638 * A058640 A058641 A058642

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

EXTENSIONS

Erroneous zero at start of sequence removed by Michael Somos, Sep 30 2009

STATUS

approved

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)