OFFSET
0,2
REFERENCES
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..301
FORMULA
a(n) = (-1)^n * Sum_{k=0..n} A060628(n,k)*4^k for n >= 0.
E.g.f. S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) S(x) = sn(x,k) at k = 2.
(2.a) S(x) = sn(2*x,1/2)/2.
(2.b) S(x) = sn(x,1/2) * cn(x,1/2) * dn(x,1/2) / (1 - sn(x,1/2)^4/4).
(3.a) S(x) = Series_Reversion( Integral 1/sqrt( (1-x^2)*(1-4*x^2) ) dx ).
(3.b) S(x) = Integral sqrt(1 - S(x)^2) * sqrt(1 - 4*S(x)^2) dx.
(4.a) S(x) = sin( Integral sqrt(1 - 4*S(x)^2) dx ).
(4.b) S(x) = sin( 2 * Integral sqrt(1 - S(x)^2) dx ) / 2.
(5.a) S(x) = sqrt(1 - cn(x,2)^2).
(5.b) S(x) = sqrt(1 - dn(x,2)^2) / 2.
O.g.f.: x/(1 + 5*x - 4*1*2^2*3*x^2/(1 + 5*3^2*x - 4*3*4^2*5*x^2/(1 + 5*5^2*x - 4*5*6^2*7*x^2/(1 + 5*7^2*x - 4*7*8^2*9*x^2/(1 + 5*9^2*x - ...))))) = x - 5*x^2 + 73*x^3 - 2765*x^4 + 171409*x^5 - 16145045*x^6 + ... (continued fraction, see Wall, 94.17, p. 374).
a(n) ~ (-1)^n * 2^(4*n+4) * agm(1,2)^(2*n+2) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - Vaclav Kotesovec, Mar 28 2024
EXAMPLE
E.g.f.: S(x) = x - 5*x^3/3! + 73*x^5/5! - 2765*x^7/7! + 171409*x^9/9! - 16145045*x^11/11! + 2168436697*x^13/13! - 391723265885*x^15/15! + ...
where S(x) = sn(x,2).
MAPLE
# a(n) = (2*n+1)! * [x^(2*n+1)] sn(x, 2).
sn_list := proc(k, len) local n; seq((2*n+1)!*coeff(series(JacobiSN(z, k), z,
2*len + 2), z, 2*n + 1), n = 0..len) end:
sn_list(2, 15); # Peter Luschny, Mar 25 2024
MATHEMATICA
nmax = 20;
DeleteCases[CoefficientList[JacobiSN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax] + 1)! (* Jean-François Alcover, Mar 28 2024 *)
PROG
(PARI) /* S(x) = Jacobi Elliptic Function sn(x, k) at k = 2: */
{a(n) = my(S, k = 2); S = serreverse( intformal( 1/sqrt((1-x^2)*(1-k^2*x^2 +x*O(x^(2*n+2)) ) ) ));
(2*n+1)!*polcoeff(S, 2*n+1)}
for(n=0, 20, print1( a(n), ", ") );
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 23 2024
STATUS
approved