A145725 McKay-Thompson series of class 60C for the Monster group with a(0) = 1.

1, 1, 1, 1, 2, 2, 2, 3, 5, 5, 5, 7, 9, 10, 11, 14, 18, 20, 22, 27, 32, 36, 40, 48, 57, 63, 70, 82, 95, 106, 119, 137, 158, 175, 195, 222, 252, 280, 311, 352, 397, 439, 486, 546, 611, 676, 747, 834, 929, 1024, 1128, 1253, 1389, 1528, 1679, 1857, 2052, 2250

Offset

-1,5

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1) * psi(-q^3) * psi(-q^5) / (psi(-q) * psi(-q^15)) in powers of q where psi() is a Ramanujan theta function.

Expansion of eta(q^2) * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60)) in powers of q.

Euler transform of period 60 sequence [ 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - u - v - u*v) * (u^3 + u^2*v + u*v^2 +v^3) + u*v * (1 + u^2) * (1 + v^2) + 2*u^2*v^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t).

G.f.: 1 / ( x * Product_{k>0} P(15, x^k) * P(60, x^k) ) where P(n, x) is the n-th cyclotomic polynomial.

a(n) = -(-1)^n * A135213(n). a(n) = A058727(n) unless n=0.

a(2*n) = A094023(n). - Michael Somos, Sep 06 2015

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Example

G.f. = 1/q + 1 + q + q^2 + 2*q^3 + 2*q^4 + 2*q^5 + 3*q^6 + 5*q^7 + 5*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(3/2)] EllipticTheta[ 2, Pi/4, q^(5/2)] / (EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(15/2)]), {q, 0, n}]; (* Michael Somos, Sep 06 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(3*k)) * (1-x^(5*k)) * (1+x^(6*k)) * (1+x^(10*k)) * (1+x^(15*k)) / ((1-x^(4*k)) * (1-x^(60*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A)), n))};

Crossrefs

Cf. A058727, A094023, A135213.

Sequence in context: A029073 A058618 A135213 * A058727 A304683 A035658

Adjacent sequences: A145722 A145723 A145724 * A145726 A145727 A145728

Keyword

nonn

Author

Michael Somos, Oct 18 2008

Status

approved