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)

1, 104, 4372, 96256, 1240002, 10698752, 74428120, 431529984, 2206741887, 10117578752, 42616961892, 166564106240, 611800208702, 2125795885056, 7040425608760, 22327393665024, 68134255043715, 200740384538624

Offset

-1,2

Comments

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

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

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from T. D. Noe)

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.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

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.

Michael Somos, Emails to N. J. A. Sloane, 1993

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

a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Apr 01 2017

Example

G.f. = 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-François Alcover, Nov 16 2015, adapted from PARI *)
nmax = 20; CoefficientList[Series[128*x + Product[1/(1 + x^k)^24, {k, 1, nmax}] + 4096*x^2*Product[(1 + x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 03 2018 *)

Prog

(PARI) {a(n) = my(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) = my(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: A187530 A185741 A299333 * A300174 A250668 A035811

Adjacent sequences: A007264 A007265 A007266 * A007268 A007269 A007270

Keyword

nonn,nice

Author

N. J. A. Sloane, Apr 28 1994

Status

approved