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.
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved