A112212 McKay-Thompson series of class 84C 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

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 first 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 q^(1/3) * eta(q^2)^2 * eta(q^14)^2 / (eta(q) * eta(q^4) * eta(q^7) * eta(q^28)) in powers of q. - Michael Somos, Dec 03 2013

Euler transform of period 28 sequence [1, -1, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 2, -1, 1, 0, 1, -1, 1, 0, ...]. - Michael Somos, Dec 03 2013

G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 03 2013

G.f.: Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(14*k - 7)). - Michael Somos, Dec 03 2013

a(n) = (-1)^n * A102314(n). a(2*n + 1) = A093950(n). - Michael Somos, Dec 03 2013

a(n) ~ exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

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 + ...
T84C = 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, Dec 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] Product[ 1 + x^k, {k, 7, n, 14}], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^14 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A)), n))}; /* Michael Somos, Dec 03 2013 */

Crossrefs

Cf. A093950, A102314.

Sequence in context: A205780 A204905 A082597 * A102314 A205146 A031248

Adjacent sequences: A112209 A112210 A112211 * A112213 A112214 A112215

Keyword

nonn

Author

Michael Somos, Aug 28 2005

Status

approved