A005797 Expansion of Jacobi nome q in terms of parameter m/16.

0, 1, 8, 84, 992, 12514, 164688, 2232200, 30920128, 435506703, 6215660600, 89668182220, 1305109502496, 19138260194422, 282441672732656, 4191287776164504, 62496081197436736, 935823746406530603, 14065763582458332888, 212122153814497767004, 3208590886304243284640

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For a faster convergent series see A002103, where k' = sqrt(1 - k^2). - Wolfdieter Lang, Jul 14 2016

The Ansatz technique of A308835, A308836, and A308837 also works to produce the coefficients of this sequence from the ODE: T-d/dx(4*(1-x)*x*dT/dx)=0. - Bradley Klee, Jul 03 2019

Examples

			G.f. = x + 8*x^2 + 84*x^3 + 992*x^4 + 12514*x^5 + 164688*x^6 + 2232200*x^7 + ...
Given g.f. A(x),  then q = exp(-Pi sqrt(6)) = A( m/16 ) where m = ((2-sqrt(3))*(sqrt(3)-sqrt(2)))^2. - _Michael Somos_, Oct 30 2019
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
• B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.
• C. L. Mallows, personal communication.
• 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..700
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
• Index entries for reversions of series

Crossrefs

Reversion of A005798. Cf. A002639. Other nomes: A308835, A308836, A308837.

Programs

Maple

a:= n-> coeff(series(EllipticNome(4*sqrt(x)), x, n+1), x, n):
seq(a(n), n=0..17);  # Thomas Richard, Aug 03 2022

Mathematica

a[ n_] := SeriesCoefficient[ EllipticNomeQ[ 16 x], {x, 0 ,n}] (* Michael Somos, Jul 11 2011 *)

PARI

{a(n) = if( n < 1, 0, polcoeff( serreverse( x * prod(k=1, n-1, (1 + x^k)^(-1)^k, 1 + x * O(x^n))^8), n))} /* Michael Somos, Jul 19 2002 */

PARI

{a(n) = my(A, m); if( n < 1, 0, m=1; A = x + O(x^2); while( m < n, m*=2; A = sqrt( subst(A, x, x^2)); A /= (1 + 4*A)^2); polcoeff( serreverse(A), n))} /* Michael Somos, Mar 18 2003 */

Formula

G.f.: q = q(m) = Sum_{n>=0} a(n) * (m/16)^n.

G.f.: exp( -Pi * agm(1, sqrt(1 - 16 * x) ) / agm(1, sqrt( 16*x ) ) ).