A112182 McKay-Thompson series of class 40d for the Monster group.

1, -1, 0, -1, 1, -2, 2, -1, 3, -3, 3, -3, 4, -5, 5, -7, 8, -8, 9, -10, 13, -15, 14, -17, 20, -23, 24, -26, 31, -34, 38, -41, 46, -52, 55, -62, 70, -75, 82, -90, 103, -112, 118, -131, 145, -161, 172, -185, 208, -225, 244, -265, 288, -316, 339, -370, 404, -435, 469, -507, 557, -601, 640, -696, 755, -818

Offset

0,6

Comments

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

References

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

Links

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

Index entries for McKay-Thompson series for Monster simple group

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of chi(-x) * chi(-x^5) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jul 02 2014

Expansion of q^(1/4) * eta(q) * eta(q^5) / (eta(q^2) * eta(q^10)) in powers of q. - Michael Somos, Jul 02 2014

Euler transform of period 10 sequence [ -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, ...]. - Michael Somos, Jul 02 2014

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (u-v^3) - 3 * u*v * (1 + u*v). - Michael Somos, Jul 02 2014

a(n) = (-1)^n * A112209(n).

a(n) ~ (-1)^n * exp(Pi*sqrt(n/5)) / (2^(3/2) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Example

G.f. = 1 - x - x^3 + x^4 - 2*x^5 + 2*x^6 - x^7 + 3*x^8 - 3*x^9 + 3*x^10 + ...
T40d = 1/q - q^3 - q^11 + q^15 - 2*q^19 + 2*q^23 - q^27 + 3*q^31 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^5, x^10], {x, 0, n}]; (* Michael Somos, Jul 02 2014 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^10 + A)), n))}; /* Michael Somos, Jul 02 2014 */

Crossrefs

Cf. A112209.

Sequence in context: A245325 A091224 A308684 * A112209 A240127 A109524

Adjacent sequences: A112179 A112180 A112181 * A112183 A112184 A112185

Keyword

sign

Author

Michael Somos, Aug 28 2005

Status

approved