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A007267 Expansion of 16 * (1 + k^2)^4 /(k * k'^2)^2 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.
(Formerly M5369)
198
1, 104, 4372, 96256, 1240002, 10698752, 74428120, 431529984, 2206741887, 10117578752, 42616961892, 166564106240, 611800208702, 2125795885056, 7040425608760, 22327393665024, 68134255043715, 200740384538624 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

REFERENCES

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195.

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 517.

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

LINKS

T. D. Noe, Table of n, a(n) for n=-1..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).

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Vol. 2.

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.

Titus Piezas III, Ramanujan's Constant exp(Pi sqrt(163)) And Its Cousins.

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Expansion of 16 * (1 + k'^2)^4 /(k' * k^2)^2 in powers of q^2. - Michael Somos, Nov 11 2006

McKay-Thompson series of class 2A for the Monster group with a(0) = 104.

EXAMPLE

1/q + 104 + 4372*q + 96256*q^2 + 1240002*q^3 + 10698752*q^4 + ...

MATHEMATICA

a[ n_] := If[ n < -1, 0, With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ 16 (1 + m)^4 /(m (1 - m)^2), {q, 0, n}]]] (* Michael Somos, Jun 29 2011 *)

a[ n_] := If[ n < -1, 0, With[ {m = ModularLambda[ Log[q]/(Pi I)]}, SeriesCoefficient[ 16 (1 + m)^4 /(m (1 - m)^2), {q, 0, n}]]] (* Michael Somos, Jun 30 2011 *)

QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = (A + 64*(q/A))^2 + O[q]^30; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 16 2015, adapted from PARI *)

PROG

(PARI) {a(n) = local(A); if( n<-1, 0, A = prod(k=1, n\2 + 1, 1 - x^(2*k - 1), 1 + x^2 * O(x^n))^12; polcoeff( (64 * x / A + A)^2, n+1))}

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( (A + 64 * x / A)^2, n))} /* Michael Somos, Nov 11 2006 */

CROSSREFS

Cf. A007241, A045478. Convolution square of A007247.

A045478, A007241, A106207, A007267, and A101558 are all essentially the same sequence.

Sequence in context: A217773 A187530 A185741 * A250668 A035811 A004393

Adjacent sequences:  A007264 A007265 A007266 * A007268 A007269 A007270

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Apr 28 1994

STATUS

approved

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Last modified February 7 22:38 EST 2016. Contains 268087 sequences.