A029841 McKay-Thompson series of class 8E for the Monster group.

1, 4, 2, -8, -1, 20, -2, -40, 3, 72, 2, -128, -4, 220, -4, -360, 5, 576, 8, -904, -8, 1384, -10, -2088, 11, 3108, 12, -4552, -15, 6592, -18, -9448, 22, 13392, 26, -18816, -29, 26216, -34, -36224, 38, 49700, 42, -67728, -51, 91688

Offset

0,2

Comments

A Hauptmodul for Gamma'_0(8).

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

References

A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 380, Section 488.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339. See page 336.

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

J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

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

G.f.: ( Product_{k>0} (1 + q^(2*k - 1)) / (1 + q^(2*k)) )^4.

Expansion of q^(1/4) * (1 + k) / k^(1/2) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus. - Michael Somos, Aug 01 2011

Expansion of q^(1/2) * 4 / k in powers of q where q is Jacobi's nome and k is the elliptic modulus. - Michael Somos, Aug 01 2011 and Feb 28 2012

Expansion of (phi(x) / psi(x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of q^(1/2) * (eta(q^2)^3 / (eta(q) * eta(q^4)^2))^4 in powers of q. - Michael Somos, Aug 01 2011

Euler transform of period 4 sequence [4, -8, 4, 0, ...]. - Michael Somos, Mar 18 2004

Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 16 + 8*v + v^2 - u^2*v. - Michael Somos, Mar 18 2004

G.f. A(q) satisfies A(q) = sqrt(A(q^2))+4*q/sqrt(A(q^2)). - Joerg Arndt, Aug 06 2011

A112143(n) = (-1)^n * a(n). a(2*n) = A029839(n). a(2*n + 1) = 4 * A079006(n). - Michael Somos, Mar 27 2004.

Convolution inverse of A001938. Convolution square of A029839. Convolution square is A029845.

Example

G.f. = 1 + 4*x + 2*x^2 - 8*x^3 - x^4 + 20*x^5 - 2*x^6 - 40*x^7 + 3*x^8 + ...
T8E = 1/q + 4*q + 2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*x^15 + ...

Mathematica

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 4 / Sqrt[m], {q, 0, n - 1/2}]]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 2 (1 + Sqrt[m]) / m^(1/4), {q, 0, n/2 - 1/4}]]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 / (QPochhammer[ x] QPochhammer[x^4]^2))^4, {x, 0, n}]; (* Michael Somos, Aug 20 2014 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A))^2)^4, n))};
(PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = (4*x + A) / sqrt(A)); polcoeff(A, n))};

Crossrefs

Cf. A001938, A029839, A029845, A079006, A112143.

Sequence in context: A241298 A019953 A241005 * A112143 A112151 A112152

Adjacent sequences: A029838 A029839 A029840 * A029842 A029843 A029844

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved