A082303 McKay-Thompson series of class 32e for the Monster group.

1, -1, -1, 0, 1, 0, -1, 1, 2, -1, -2, 1, 2, -1, -3, 1, 4, -2, -5, 2, 5, -2, -6, 3, 8, -4, -9, 4, 10, -4, -12, 6, 15, -7, -17, 7, 19, -8, -22, 10, 26, -12, -30, 13, 33, -14, -38, 17, 45, -21, -51, 22, 56, -24, -64, 29, 74, -33, -83, 36, 92, -40, -104, 46, 119, -53, -133, 58

Offset

0,9

Comments

Number 4 of the 130 identities listed in Slater 1952. - Michael Somos, Aug 21 2015

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

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. see page 274.

Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp.147-167, (1952).

Index entries for McKay-Thompson series for Monster simple group

Formula

Euler transform of period 4 sequence [ -1, -1, -1, 0, ...].

Expansion of q^(1/8) * eta(q) / eta(q^4) in powers of q.

Given g.f. A(x), then B(q) = (A(q^8) / q)^8 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v + 16) * (u + 16) * u - v^2. - Michael Somos, Jan 09 2005

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

a(n) = (-1)^n * A029838(n).

Convolution square is A082304.

G.f.: 2 - 2/(1+Q(0)), where Q(k)= 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 02 2013

G.f.: Sum_{k>=0} (-1)^k * q^k^2 * Product_{i=1..k} (1 + x^(2*i - 1)) / (1 - x^(4*i)). - Michael Somos, Aug 21 2015

a(n) = -(1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

abs(a(n)) ~ sqrt(sqrt(2) + (-1)^n) * exp(Pi*sqrt(n)/2^(3/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Feb 07 2023

Example

G.f. = 1 - x - x^2 + x^4 - x^6 + x^7 + 2*x^8 - x^9 - 2*x^10 + x^11 + 2*x^12 + ...
T32e = 1/q - q^7 - q^15 + q^31 - q^47 + q^55 + 2*q^63 - q^71 - 2*q^79 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^4], {x, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ -x^2, x^2], {x, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (16 (1 - m)/m)^(1/8), {q, 0, n - 1/8}]]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}] / Product[ 1 + x^k, {k, 2, n, 2}], {x, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-x}, {-x^2}, x^2, x], {x, 0, n}]; (* Michael Somos, Aug 21 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^k^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x^4, x^4, k], {k, 0, Sqrt@n}], {x, 0, n}]]; (* Michael Somos, Aug 21 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^4 + A), n))};
(PARI) q='q+O('q^66); Vec( eta(q)/eta(q^4) ) \\ Joerg Arndt, Mar 25 2017

Crossrefs

Cf. A029838, A082304.

Sequence in context: A085342 A025825 A293224 * A316384 A029838 A213649

Adjacent sequences: A082300 A082301 A082302 * A082304 A082305 A082306

Keyword

sign

Author

Michael Somos, Apr 08 2003

Status

approved