A187020 McKay-Thompson series of class 36D for the Monster group with a(0) = 1.

1, 1, 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

-1,3

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = -1..1000

K. Bringmann and H. Swisher, On a conjecture of Koike on identities between Thompson series and Rogers-Ramanujan functions, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2317-2326. (MR230255) see p. 2325 Appendix A.

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

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 (1/q) * chi(-q^9) * chi(-q^18) / (chi(-q) * chi(-q^2)) in powers of q where chi() is a Ramanujan theta function.

Expansion of eta(q^4) * eta(q^9) / (eta(q) * eta(q^36)) in powers of q.

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

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

G.f. A(x) = ( G(x^36) * H(x) - x^7 * H(x^36) * G(x) ) / ( G(x^9) * H(x^4) - x * H(x^9) * G(x^4) ) where G(x) (g.f. of A003114) and H(x) (g.f. of A003106) are Rogers-Ramanujan functions [Bringmann + Swisher, (A.12)]

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) = A058647(n) = A186964(n) unless n=0. A128128(n) = a(2*n). A000005(n) = a(2*n - 1).

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

Example

1/q + 1 + 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^(4*k)) * (1-x^(9*k)) / ((1-x^k) * (1-x^(36*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
QP = QPochhammer; s = QP[q^4]*(QP[q^9]/(QP[q]*QP[q^36])) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)

Prog

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

Crossrefs

Cf. A000005, A003106, A003114, A058647, A128128, A186964.

Sequence in context: A155510 A073576 A186115 * A058647 A305630 A371178

Adjacent sequences: A187017 A187018 A187019 * A187021 A187022 A187023

Keyword

nonn

Author

Michael Somos, Mar 06 2011

Status

approved