A132107 -id:A132107 - cp's OEIS frontend

A007262 McKay-Thompson series of class 6c for Monster.

1, -6, 9, 16, -66, 54, 98, -300, 243, 364, -1128, 828, 1221, -3498, 2511, 3528, -9876, 6804, 9358, -25428, 17217, 23068, -61644, 40824, 53916, -141318, 92340, 119912, -310554, 199980, 256792, -656436, 418311, 530960, -1344144, 847584, 1066157, -2673372, 1671741, 2084464, -5186118, 3216834, 3981926, -9832752, 6057504, 7445924, -18269124, 11181636, 13661725, -33315852, 20274948, 24630344, -59740716

Offset: 0

N. J. A. Sloane

sign

			T6c = 1/q - 6*q + 9*q^3 + 16*q^5 - 66*q^7 + 54*q^9 + 98*q^11 - 300*q^13 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
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).
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.
Index entries for McKay-Thompson series for Monster simple group

Cf. A132107.

eta[q_] := q^(1/24)*QP0chhammer[q]; a[n_]:= SeriesCoefficient[ q^(-1)*(eta[q^2]/ eta[q^6])^6, {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 25 2018 *)

N=66; q='q+O('q^N); Vec( (eta(q^1)/eta(q^3))^6/q ) \\ Joerg Arndt, Apr 09 2016

G.f.: (E(q^2)/E(q^6))^6 / q where E(q) = Product_{n>=1} (1 - q^n); note that every second term is zero and is omitted in this sequence, cf. the PARI/GP code. - Joerg Arndt, Apr 09 2016

More terms from Joerg Arndt, Apr 09 2016

A264026 Expansion of (f(x^3) / f(x))^6 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 27, -92, 279, -756, 1913, -4536, 10260, -22220, 46479, -94176, 185749, -357426, 673056, -1242404, 2252772, -4017816, 7058609, -12228060, 20911230, -35330324, 59023728, -97568712, 159693831, -258941124, 416181510, -663337512, 1048935414, -1646245836

Offset: 0

Michael Somos, Nov 01 2015

sign

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

			G.f. = 1 - 6*x + 27*x^2 - 92*x^3 + 279*x^4 - 756*x^5 + 1913*x^6 - 4536*x^7 + ...
G.f. = q - 6*q^3 + 27*q^5 - 92*q^7 + 279*q^9 - 756*q^11 + 1913*q^13 - 4536*q^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000726, A132107.

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3] / QPochhammer[ -x])^6, {x, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/2)* (eta[q]*eta[q^4]*eta[q^6]^3/(eta[q^2]^3*eta[q^3]*eta[q^12]))^6, {q, 0, 60}], q]; Table[a[[n]], {n,1,50}] (* G. C. Greubel, Jul 04 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A)))^6, n))};

Expansion of q^(-1/2) * (eta(q) * eta(q^4) * eta(q^6)^3 / (eta(q^2)^3 * eta(q^3) * eta(q^12)))^6 in powers of q.
Euler transform of period 12 sequence [ -6, 12, 0, 6, -6, 0, -6, 6, 0, 12, -6, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132107.
Convolution inverse of A132107.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(13/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

cp's OEIS Frontend

A007262 McKay-Thompson series of class 6c for Monster.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions