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A005797 Expansion of Jacobi nome q in terms of parameter m/16.
(Formerly M4561)
13

%I M4561

%S 0,1,8,84,992,12514,164688,2232200,30920128,435506703,6215660600,

%T 89668182220,1305109502496,19138260194422,282441672732656,

%U 4191287776164504,62496081197436736,935823746406530603

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

%C For a faster convergent series see A002103, where k' = sqrt(1 - k^2). - _Wolfdieter Lang_, Jul 14 2016

%C 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

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.

%D 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.

%D C. L. Mallows, personal communication.

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

%H Vaclav Kotesovec, <a href="/A005797/b005797.txt">Table of n, a(n) for n = 0..700</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

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

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

%e G.f. = x + 8*x^2 + 84*x^3 + 992*x^4 + 12514*x^5 + 164688*x^6 + 2232200*x^7 + ...

%e 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

%t a[ n_] := SeriesCoefficient[ EllipticNomeQ[ 16 x], {x, 0 ,n}] (* _Michael Somos_, Jul 11 2011 *)

%o (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 */

%o (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 */

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

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

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Last modified July 9 22:46 EDT 2020. Contains 335570 sequences. (Running on oeis4.)