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A007253 McKay-Thompson series of class 5a for Monster.
(Formerly M4131)
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,3

COMMENTS

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

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)

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

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

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

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Last modified January 29 16:50 EST 2023. Contains 359923 sequences. (Running on oeis4.)