A227585 McKay-Thompson series of class 36A for the Monster group with a(0) = 2.

1, 2, 3, 2, 3, 6, 10, 12, 15, 22, 30, 36, 44, 60, 78, 96, 117, 150, 190, 228, 276, 340, 420, 504, 603, 732, 885, 1052, 1245, 1488, 1770, 2088, 2454, 2902, 3420, 3996, 4666, 5460, 6378, 7400, 8583, 9972, 11566, 13344, 15378, 17752, 20448, 23472, 26904, 30876

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of 3 + psi(-q) / (q * psi(-q^9)) + 3 * q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.

Expansion of (1/q) * (psi(-q^3)^2 / (psi(-q) * psi(-q^9)))^2 in powers of q where psi() is a Ramanujan theta function.

Expansion of -3 * b(-q) * c(-q) * (b(q^9) / (b(q^2) * c(q^2) * b(-q^3)))^2 in powers of q where b(), c() are cubic AGM theta functions.

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

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

a(n) = -(-1)^n * A215412(n). a(n) = A058644(n) unless n=0.

Convolution square of A112205.

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2015

Example

1/q + 2 + 3*q + 2*q^2 + 3*q^3 + 6*q^4 + 10*q^5 + 12*q^6 + 15*q^7 + 22*q^8 + ...

Mathematica

nmax = 60; CoefficientList[Series[Product[((1+x^k) * (1-x^(3*k))^2 * (1+x^(6*k))^2 * (1+x^(9*k)) / ((1-x^(4*k)) * (1-x^(36*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2015 *)
a[n_]:= SeriesCoefficient[3 - EllipticTheta[2, 0, I*q^(1/2)]/EllipticTheta[2, 0, I*q^(9/2)] - 3*EllipticTheta[2, 0, I*q^(9/2)]/EllipticTheta[2, 0, q^(1/2)], {q, 0, n}]; Table[a[n], {n, -1, 50}] (* G. C. Greubel, Feb 18 2018 *)

Prog

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

Crossrefs

Cf. A058644, A112205, A215412.

Sequence in context: A193917 A089135 A215412 * A038063 A264506 A085204

Adjacent sequences: A227582 A227583 A227584 * A227586 A227587 A227588

Keyword

nonn

Author

Michael Somos, Jul 16 2013

Status

approved