A210459 McKay-Thompson series of class 20A for the Monster group with a(0) = 4.

1, 4, 6, 8, 17, 32, 54, 80, 116, 192, 290, 408, 585, 832, 1192, 1648, 2237, 3072, 4156, 5576, 7414, 9824, 12964, 16896, 22002, 28544, 36794, 47184, 60185, 76736, 97388, 122864, 154615, 194048, 242904, 302800, 376271, 466720, 577176, 711840, 875611, 1074752

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 q^-1 * (chi(q) * chi(q^5))^4 in powers of q where chi() is a Ramanujan theta function.

Euler transform of period 20 sequence [ 4, -4, 4, 0, 8, -4, 4, 0, 4, -8, 4, 0, 4, -4, 8, 0, 4, -4, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = f(t) where q = exp(2 Pi i t).

G.f.: (Product_{k>0} (1 + (-x)^k) * (1 + (-x)^(5*k)))^-4.

a(n) = A112158(n) unless n=0. a(n) = 5*A210458(n) + A145740(n) unless n=0. A132040(n) = (-1)^n * a(n). Convolution square of A112179.

a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

Example

G.f. = 1/q + 4 + 6*q + 8*q^2 + 17*q^3 + 32*q^4 + 54*q^5 + 80*q^6 + 116*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q^2] QPochhammer[ -q^5, q^10])^4, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
nmax = 50; CoefficientList[Series[Product[((1 + x^(2*k-1))/((1 + x^(10*k))*(1 - x^(10*k-5))))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 30 2017 *)

Prog

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

Crossrefs

Cf. A112158, A112179, A132040, A145740, A210458.

Sequence in context: A022599 A112160 A132040 * A346868 A279896 A247280

Adjacent sequences: A210456 A210457 A210458 * A210460 A210461 A210462

Keyword

nonn

Author

Michael Somos, Jan 21 2013

Status

approved