A132320 McKay-Thompson series of class 22B for the Monster group with a(0) = -2.

1, -2, 1, -2, 4, -4, 5, -6, 9, -12, 13, -18, 25, -28, 33, -44, 54, -64, 74, -92, 114, -132, 155, -186, 224, -260, 303, -360, 424, -488, 565, -662, 770, -888, 1018, -1180, 1366, -1560, 1780, -2048, 2345, -2668, 3034, -3460, 3946, -4468, 5052, -5734, 6502, -7328

Offset

-1,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = -1..5000

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^11))^2 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q) * eta(q^11) / (eta(q^2) * eta(q^22)))^2 in powers of q.

Euler transform of period 22 sequence [ -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -4, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (22 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123631.

G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(11*k)))^-2.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 4 * u + 4 * u * v.

A = A058568(n) unless n = 0. Convolution inverse is A123631.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/11)) / (2*11^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

Example

G.f. = 1/q - 2 + q - 2*q^2 + 4*q^3 - 4*q^4 + 5*q^5 - 6*q^6 + 9*q^7 - ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q, q^2] QPochhammer[ q^11, q^22])^2 / q, {q, 0, n}]; (* Michael Somos, Aug 27 2014 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A) / (eta(x^2 + A) * eta(x^22 + A)))^2, n))};

Crossrefs

Cf. A058568, A123631.

Sequence in context: A022597 A073252 A134005 * A353400 A076369 A328790

Adjacent sequences: A132317 A132318 A132319 * A132321 A132322 A132323

Keyword

sign

Author

Michael Somos, Aug 18 2007

Status

approved