A000706 - OEIS

%I M5458 N2367 #29 Nov 08 2015 05:24:17

%S 1,504,270648,144912096,77599626552,41553943041744,22251789971649504,

%T 11915647845248387520,6380729991419236488504,

%U 3416827666558895485479576,1829682703808504464920468048,979779820147442370107345764512

%N Expansion of modular function 1/E_3 (cf. A013973).

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

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

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

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

%H Vaclav Kotesovec, <a href="/A000706/b000706.txt">Table of n, a(n) for n = 0..360</a>

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper37/page1.htm">On the coefficients in the expansions of certain modular functions</a>, 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

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

%F 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

%F Convolution inverse of A013973.

%F 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

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

%t a[ n_] := SeriesCoefficient[ 1 / (1 + Sum[ -504 DivisorSigma[ 5, k] q^k, {k, n}]), {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t 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 *)

%t 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 *)

%t 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 *)

%o (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 */

%o (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 */

%Y Cf. A013973, A259150.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_