A007253 McKay-Thompson series of class 5a for Monster. (Formerly M4131)

1, 0, -6, 20, 15, 36, 0, -84, 195, 100, 240, 0, -461, 1020, 540, 1144, 0, -1980, 4170, 2040, 4275, 0, -6984, 14340, 6940, 14076, 0, -21936, 44025, 20760, 41476, 0, -62484, 123620, 57630, 113244, 0, -166056, 324120, 148900, 289578, 0

Offset

-1,3

References

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

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000 (terms -1..100 from G. A. Edgar)

Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 8.

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).

John McKay and Hubertus 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

Formula

G.f.: T5a(q) satisfies functional equation P5(T5a(q)) = j(q^5) - 744, where we used modular function j(q) from A000521 and polynomial P5(t) = t^5+30*t^3-100*t^2+105*t-780. - G. A. Edgar, Mar 10 2017

G.f. is a period 1 Fourier series which satisfies f(-1 / (25 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 23 2023

G.f.: A(q) satisfies functional equation P(A(q)) = j(q^5), where P(x) = (x-1)^3 * (x^2 + 3*x + 36) and j is Klein's modular function. - Michael Somos, Jan 23 2023

Example

T5a = 1/q - 6*q + 20*q^2 + 15*q^3 + 36*q^4 - 84*q^6 + 195*q^7 + 100*q^8 + ...

Maple

with(numtheory): TOP := 23;
Order:=101;
g2 := (4/3) * (1 + 240 * add(sigma[ 3 ](n)*q^n, n=1..TOP-1));
g3 := (8/27) * (1 - 504 * add(sigma[ 5 ](n)*q^n, n=1..TOP-1));
delta := series(g2^3 - 27*g3^2, q=0, TOP);
j := series(1728 * g2^3 / delta, q=0, TOP);
# computation above of j is from A000521
P5 := t^5 + 30*t^3 - 100*t^2 + 105*t - 780;
subs(t = q^(-1) + x, P5) - subs(q=q^5, j - 744);
solve(%, x);
T5a := series(q^(-1)+%, q=0) assuming q > 0;
# G. A. Edgar, Mar 10 2017

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; e5B:= (eta[q]/eta[q^5])^6; e25a:= (eta[q]/eta[q^25]);  a[n_]:= SeriesCoefficient[(1 + 5/e25a)*(1 + e5B) + 5*(e25a - 5/e25a)*(e5B/(e25a)^3), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* G. C. Greubel, Jan 25 2018 *)

Prog

(PARI) q='q+O('q^30);  F=(1 + 5*q*eta(q^25)/eta(q))*(1 + (eta(q)/eta(q^5) )^6/q) + 5*(eta(q)/(q*eta(q^25)) - 5*q*eta(q^25)/eta(q))*(q^2* eta(q^25)^3 *eta(q)^3/eta(q^5)^6); Vec(F)  \\ G. C. Greubel, Jun 12 2018

Crossrefs

Cf. A000521.

Sequence in context: A087998 A096823 A321328 * A096897 A063601 A222604

Adjacent sequences: A007250 A007251 A007252 * A007254 A007255 A007256

Keyword

sign,changed

Author

N. J. A. Sloane

Extensions

More terms from G. A. Edgar, Mar 10 2017

Status

approved