A102314 McKay-Thompson series of class 42C for the Monster group.

1, -1, 0, -1, 1, -1, 1, -2, 3, -2, 3, -3, 4, -4, 4, -6, 7, -7, 7, -9, 10, -12, 13, -14, 17, -18, 19, -22, 26, -28, 29, -34, 38, -41, 44, -50, 57, -60, 65, -72, 81, -86, 94, -105, 114, -124, 133, -146, 161, -174, 187, -204, 224, -240, 258, -282, 309, -332, 354, -386, 419, -450, 481, -524, 569, -606, 651, -703

Offset

0,8

Comments

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

Given g.f. A(x), the second term of the left side of Cayley's identity is -A(q). - Michael Somos, Dec 03 2013

References

A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of chi(-x) * chi(-x^7) in powers of x where chi() is a Ramanujan theta function.

Expansion of q^(1/3) * eta(q) * eta(q^7) / (eta(q^2) * eta(q^14)) in powers of q.

Euler transform of period 14 sequence [ -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, ...].

Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^2*v - 2*u.

G.f. is a period 1 Fourier series which satisfies f(-1 / (126 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A093950.

G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^(7*k))).

a(n) = (-1)^n * A112212(n). a(2*n + 1) = - A093950(n). a(4*n) = A193826(n). a(4*n + 2) = A193883(n).

Convolution inverse is A093950.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - 2*x^7 + 3*x^8 - 2*x^9 + 3*x^10 - 3*x^11 + ...
T42C = 1/q - q^2 - q^8 + q^11 - q^14 + q^17 - 2*q^20 + 3*q^23 - 2*q^26 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, n}]; (* Michael Somos, Aug 06 2011 *)
a[ n_] := SeriesCoefficient[ 1 / ( Product[ 1 + x^k, {k, n}] Product[ 1 + x^k, {k, 7, n, 7}] ), {x, 0, n}]; (* Michael Somos, Aug 06 2011 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)), n))};

Crossrefs

Cf. A093950, A112212, A193826, A193883.

Sequence in context: A204905 A082597 A112212 * A205146 A031248 A030582

Adjacent sequences: A102311 A102312 A102313 * A102315 A102316 A102317

Keyword

sign

Author

Michael Somos, Jan 03 2005

Status

approved