A005797 Expansion of Jacobi nome q in terms of parameter m/16. (Formerly M4561)

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

Offset

0,3

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

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

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 ) ) ).

Example

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

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A130591 A166483 A048665 * A233835 A300993 A052659

Adjacent sequences: A005794 A005795 A005796 * A005798 A005799 A005800

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved