A000706 Expansion of modular function 1/E_3 (cf. A013973). (Formerly M5458 N2367)

1, 504, 270648, 144912096, 77599626552, 41553943041744, 22251789971649504, 11915647845248387520, 6380729991419236488504, 3416827666558895485479576, 1829682703808504464920468048, 979779820147442370107345764512

Offset

0,2

Comments

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

References

S. Ramanujan, Collected Papers of Srinivasa Ramanujan, pp. 115-7, Ed. G. H. Hardy et al., AMS Chelsea 2000, p. 317.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..360

S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155 [G. H. Hardy, Coll. Papers, Vol. 1, 294-305.] - Added by N. J. A. Sloane, Feb 21 2010

Formula

Expansion of 1 / R(q) in powers of q where R() is a Ramanujan Lambert series.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2*w^2 + 121*u^2*w^2 + 4096*u^2*v^2 - 8*v^3*w - 512*u*v^3 - 66*u*v*w^2 + 592*u*v^2*w - 4224*u^2*v*w. - Michael Somos, Aug 09 2007

Convolution inverse of A013973.

Asymptotics [Ramanujan]: a(n) ~ c * exp(2*Pi*n), where c = 2 / (96^2 * exp(-8*Pi/3) * Product_{j>=1} (1-exp(-4*Pi*j))^16) = 8192 * Pi^12 / (9 * Gamma(1/4)^16) = 0.943732053240742502013763912292610373458373085328537967959184338319972... . - Vaclav Kotesovec, Nov 08 2015

Example

G.f. = 1 + 504*q + 270648*q^2 + 144912096*q^3 + 77599626552*q^4 + 41553943041744*q^5 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1 / (1 + Sum[ -504 DivisorSigma[ 5, k] q^k, {k, n}]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, 1 / (t2^3 - 33 (t2 + t3) t2 t3 + t3^3)], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t3 = EllipticTheta[ 3, 0, q]^4, t4 = EllipticTheta[ 4, 0, q]^4}, 2 / (t3^3 - 3 (t3 - t4)^2 (t3 + t4) + t4^3) ], {q, 0, 2 n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {e1 = QPochhammer[ q]^8, e4 = 32 q QPochhammer[ q^4]^8}, QPochhammer[ q^2]^12 / ((e1 + e4) (e1^2 - 16 e1 e4 - 8 e4^2)) ], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / sum(k=1, n, -504*sigma(k, 5)*x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Aug 09 2007 */
(PARI) {a(n) = my(A, e1, e4); if( n<0, 0, A = x * O(x^n); e1 = eta(x + A)^8; e4 = 32 * x * eta(x^4 + A)^8; polcoeff( eta(x^2 + A)^12 / ((e1 + e4) * (e1^2 - 16*e1*e4 - 8*e4^2)), n))}; /* Michael Somos, Apr 26 2015 */

Crossrefs

Cf. A013973, A259150.

Sequence in context: A288851 A105097 A278308 * A289637 A226266 A377222

Adjacent sequences: A000703 A000704 A000705 * A000707 A000708 A000709

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved