A136560 Coefficients of replicable function number 49a with a(0) = 3.

1, 3, 2, 1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 19, 25, 28, 35, 41, 50, 58, 71, 81, 98, 113, 134, 154, 183, 208, 244, 280, 326, 371, 431, 489, 565, 641, 735, 832, 953, 1075, 1225, 1382, 1569, 1764, 1999, 2243, 2533, 2839, 3195, 3575, 4018, 4484, 5026, 5604, 6267, 6975

Offset

-1,2

Comments

Denoted 49Z in Conway+Norton with a slight typo in the formula on page 337.

Links

Vaclav Kotesovec, Table of n, a(n) for n = -1..9998

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

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

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of ( eta(q^7)^4 + 7 * eta(q)^2 * eta(q^49)^2 ) / ( eta(q) * eta(q^49) * ( eta(q)^2 + 7 * eta(q) * eta(q^49) + 7 * eta(q^49)^2 ) ) in powers of q.

a(n) ~ exp(4*Pi*sqrt(n)/7) / (sqrt(14)*n^(3/4)). - Vaclav Kotesovec, Dec 04 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 06 2018

Example

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

Mathematica

QP = QPochhammer; A1 = QP[q]; A2 = QP[q^7]; A3 = QP[q^49]; s = (A2^4 + 7*q^3*A1^2*A3^2)/(A1*A3)/(A1^2 + 7*q^2*A1*A3 + 7*q^4*A3^2) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
a[ n_] := With[ {e1 = QPochhammer[ q], e2 = QPochhammer[ q^7], e3 = QPochhammer[ q^49]}, SeriesCoefficient[ (e2^4 + 7 q^3 e1^2 e3^2) / (q e1 e3 (e1^2 + 7 q^2 e1 e3 + 7 q^4 e3^2)), {q, 0, n}]]; (* Michael Somos, Sep 06 2018 *)

Prog

(PARI) {a(n) = my(A, A1, A2, A3); if( n<-1, 0, n++; A = x * O(x^n); A1 = eta(x + A); A2 = eta(x^7 + A); A3 = eta(x^49 + A); polcoeff( (A2^4 + 7 * x^3 * A1^2 * A3^2) / (A1 * A3) / (A1^2 + 7 * x^2 * A1*A3 + 7 * x^4 * A3^2 ), n))};

Crossrefs

Cf. A058700(n) = a(n) unless n=0.

Sequence in context: A120441 A232191 A183041 * A105847 A240485 A048984

Adjacent sequences: A136557 A136558 A136559 * A136561 A136562 A136563

Keyword

nonn

Author

Michael Somos, Jan 05 2008

Status

approved