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A014103 Expansion of (eta(q^2) / eta(q))^24 in powers of q. 2
1, 24, 300, 2624, 18126, 105504, 538296, 2471424, 10400997, 40674128, 149343012, 519045888, 1718732998, 5451292992, 16633756008, 49010118656, 139877936370, 387749049720, 1046413709980, 2754808758144, 7087483527072, 17848133716832 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Given g.f. A(q), Greenhill (1895) denotes -64 * A(q^2) by tau_0 on page 409 equation (43). - Michael Somos, Jul 17 2013

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

A. G. Greenhill, The Transformation and Division of Elliptic Functions, Proceedings of the London Mathematical Society (1895) 403-486.

LINKS

Table of n, a(n) for n=1..22.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for reversions of series

R. S. Maier, On Rationally Parametrized Modular Equations see page 4 equation (4)

FORMULA

REVERT(A005149).

Euler transform of period 2 sequence [24, 0, 24, 0, ...]. - Michael Somos Mar 19 2004

Expansion of (lambda / 16)^2 / (1 - lambda) in powers of q = exp(2 pi i t). - Michael Somos Nov 19 2005

Expansion of q / chi(-q)^24 in powers of q where chi() is a Ramanujan theta function.

Expansion of (theta_2(q) * theta_3(q) / (2 * theta_4(q)^2))^4 = (theta_2(q^(1/2))^2 / (4*theta_4(q^(1/2)) * theta_3(q^(1/2))))^4 in powers of q.

G.f.: x * Product_{k>0} (1 + x^k)^24 = x / Product_{k>0} (1 - x^(2*k - 1))^24.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 48*u*v - 4096*u*v^2. - Michael Somos Mar 19 2004

G.f. is a Fourier series which satisfies f(-1 / (2 t)) = (1/4096) / f(t) where q = exp(2 pi i t). - Michael Somos Aug 19 2007

j(q) = (f(q) + 16)^3 / f(q), j(q^2) = (f(q) + 256)^3 / f(q)^2 where j(q) is g.f. for A000521 and f(q) is 4096 times g.f. a(n). - Michael Somos Oct 01 2007

Convolution inverse of A007191. Series reversion of A005149.

Empirical : sum(exp(-2*Pi)^n*a(n),n=1..infinity) = 1/512. Simon Plouffe, Feb. 20, 2011.

EXAMPLE

q + 24*q^2 + 300*q^3 + 2624*q^4 + 18126*q^5 + 105504*q^6 + 538296*q^7 + ...

MAPLE

q*mul((1+q^m)^24, m=1..30);

MATHEMATICA

a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^-24 , {q, 0, n}] (* Michael Somos Jul 11 2011 *)

a[ n_] := SeriesCoefficient[ q / Product[ 1 - q^k, {k, 1, n + 1, 2}]^24 , {q, 0, n}] (* Michael Somos Jul 11 2011 *)

a[ n_] := With[ {m = ModularLambda[ Log[q]/(Pi I)]}, SeriesCoefficient[ (m/16)^2 / (1 - m), {q, 0, 2 n}]] (* Michael Somos Jul 11 2011 *)

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m/16)^2 /(1 - m), {q, 0, 2 n}]] (* Michael Somos Jul 11 2011 *)

PROG

(PARI) {a(n) = polcoeff( x * prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^24, n)}

(PARI) {a(n) = local(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst( A, x, x^2); A2 = A * (1 + 16*A); A = 8 * A2 + (1 + 32*A) * sqrt(A2)); polcoeff( A + 16 * A^2, n))}

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^24, n))}

CROSSREFS

Cf. A005149, A007191.

Sequence in context: A162686 A010976 A100130 * A206002 A000552 A233876

Adjacent sequences:  A014100 A014101 A014102 * A014104 A014105 A014106

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Michael Somos, Nov 24, 2001

STATUS

approved

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Last modified April 23 13:33 EDT 2014. Contains 240928 sequences.