A212253 McKay-Thompson series of class 35B for the Monster group with a(0) = 1.

1, 1, 2, 3, 5, 6, 10, 12, 18, 23, 31, 39, 54, 66, 86, 107, 137, 168, 213, 259, 323, 392, 482, 580, 711, 850, 1029, 1228, 1476, 1750, 2093, 2470, 2934, 3453, 4078, 4780, 5625, 6566, 7689, 8952, 10440, 12113, 14080, 16286, 18865, 21764, 25127, 28910, 33289

Offset

-1,3

Comments

Fricke denotes the g.f. by tau(omega) = z0/z1 on page 445.

References

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 445. Eqs. (22), (26)

Links

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

Formula

Expansion of eta(q^5) * eta(q^7) / (eta(q) * eta(q^35)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (35 t)) = f(t) where q = exp(2 Pi i t).

G.f.: (1/x) * Product_{k>0} (1 - x^(5*k)) * (1 - x^(7*k)) / ((1 - x^k) * (1 - x^(35*k))).

a(n) = A058641(n) unless n = 0.

a(n) ~ exp(4*Pi*sqrt(n/35)) / (sqrt(2) * 35^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

Example

G.f. = 1/q + 1 + 2*q + 3*q^2 + 5*q^3 + 6*q^4 + 10*q^5 + 12*q^6 + 18*q^7 + 23*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q^5] QPochhammer[ q^7] / (QPochhammer[ q] QPochhammer[ q^35]), {q, 0, n}]; (* Michael Somos, Apr 25 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^5 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^35 + A)), n))};

Crossrefs

Cf. A058641.

Sequence in context: A097071 A105420 A058641 * A237831 A347443 A329235

Adjacent sequences: A212250 A212251 A212252 * A212254 A212255 A212256

Keyword

nonn

Author

Michael Somos, Jun 28 2012

Status

approved