A186964 McKay-Thompson series of class 36D for the Monster group with a(0) = 2.

1, 2, 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

Offset

-1,2

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).

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * psi(q) * psi(q^9) / (psi(-q) * chi(q^3) * psi(q^18)) in power of q where psi(), chi() are Ramanujan theta functions.

Expansion of eta(q^2)^3 * eta(q^3) * eta(q^12) * eta(q^18)^3 / (eta(q)^2 * eta(q^4) * eta(q^6)^2 * eta(q^9) * eta(q^36)^2) in powers of q.

Euler transform of period 36 sequence [ 2, -1, 1, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 1, 0, 2, -2, 2, 0, 1, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 1, -1, 2, 0, ...].

G.f. is a period 1 Fourier series that satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058647(n) = A187020(n) unless n=0. [typo corrected by Vaclav Kotesovec, Oct 10 2015]

a(n) ~ exp(2*sqrt(n)*Pi/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 10 2015

Example

1/q + 2 + 2*q + 3*q^2 + 4*q^3 + 6*q^4 + 9*q^5 + 12*q^6 + 16*q^7 + 21*q^8 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1-x^k) * (1+x^(6*k)) * (1+x^(9*k)) / ((1-x^(4*k)) * (1+x^(3*k)) * (1+x^(18*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 10 2015 *)
a[n_]:= SeriesCoefficient[(-q)^(1/8)*EllipticTheta[2, 0, Sqrt[q]]* EllipticTheta[2, 0, Sqrt[q^9]]/(QPochhammer[-q^3, q^6]*EllipticTheta[2, 0, I*Sqrt[q]]*EllipticTheta[2, 0, Sqrt[q^18]]), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* G. C. Greubel, Jan 02 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)^3 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A)^2), n))}

Crossrefs

Cf. A058647, A187020.

Sequence in context: A163770 A035561 A068106 * A005856 A157876 A289433

Adjacent sequences: A186961 A186962 A186963 * A186965 A186966 A186967

Keyword

nonn

Author

Michael Somos, Mar 06 2011

Status

approved