A159600 E.g.f. C(x) satisfies: C(x) = (1 - 2*S(x)^2)^(1/4), where S'(x) = C(x)^3 and C'(x) = -S(x) with C(0)=1.

1, -1, 3, -27, 441, -11529, 442827, -23444883, 1636819569, -145703137041, 16106380394643, -2164638920874507, 347592265948756521, -65724760945840254489, 14454276753061349098587

Offset

0,3

Comments

See A104203 for the expansion of the sine lemniscate function sl(x).

E.g.f. C(x) is an even function; zero terms are omitted.

Radius of convergence is |x| <= r:

r = sqrt(2)*(Pi/2)^(3/2)/gamma(3/4)^2 with

C(r) = gamma(3/4)^2/(Pi/2)^(3/2) where:

r = L/sqrt(2) where L=Lemniscate constant;

r = 1.8540746773013719184338503471952600...

C(r) = 0.76275976350181318806232598096361579...

Links

Table of n, a(n) for n=0..14.

Formula

E.g.f. C(x) satisfies: C(x)^4 + 2*S(x)^2 = 1 where S(x) = Integral [1 - 2*S(x)^2]^(3/4) dx with S(0)=0; Left-shift of the Laplace transform of e.g.f. C(x) equals the Laplace transform of S(x).

E.g.f.: Sum_{k>=0} 2^k * a(k) * x^(2*k) / (2*k)! = cos lemn(x) where cos lemn(x) is the cosine lemniscate function of Gauss. - Michael Somos, Apr 25 2011

O.g.f.: 1/(1 + 1^2*x/(1 + 2^2/2*x/(1 + 3^2*x/(1 + 4^2/2*x/(1 + 5^2*x/(1 + 6^2/2*x/(1 + 7^2*x/(1 + 8^2/2*x/(1+...))))))))) (continued fraction). - Paul D. Hanna, Jul 29 2011

a(n) = (-1)^floor(n/2) * A193544(n) = (-1)^ceiling(n/2) * A193544(n) = -A159601(n). - M. F. Hasler, Aug 31 2012

G.f.: 1/Q(0) where p=1/2, Q(k) = 1 + x*(2*k+1)^2/( 1 + p*x*(2*k+2)^2/Q(k+1) ); (continued fraction due to Stieltjes T.J.). - Sergei N. Gladkovskii, Mar 22 2013

From Paul D. Hanna, Jun 02 2015: (Start)

E.g.f. C(x) satisfies:

(1) C(x) = exp( Integral C(x) * Integral -1/C(x)^3 dx dx ).

(2) C(x) = exp( Integral -1/C(x) * Integral C(x)^3 dx dx ).

(End)

G.f.: 1 / (1 + b(1)*x / (1 + b(2)*x / (1 + b(3)*x / ... ))) where b = A129194. - Michael Somos, Jan 03 2013

Example

E.g.f.: C(x) = 1 - x^2/2! + 3*x^4/4! - 27*x^6/6! + 441*x^8/8! -+ ...
C(x)^2 = 1 - 2*x^2/2! + 12*x^4/4! - 144*x^6/6! + 3024*x^8/8! -+ ...
C(x)^3 = 1 - 3*x^2/2! + 27*x^4/4! - 441*x^6/6! + 11529*x^8/8! -+ ...
C(x)^4 = 1 - 4*x^2/2! + 48*x^4/4! - 1008*x^6/6! + 32256*x^8/8! -+ ...
C(x)^4 + 2*S(x)^2 = 1 where:
S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! + ...
S(x)^2 = 2*x^2/2! - 24*x^4/4! + 504*x^6/6! - 16128*x^8/8! +-...
From Paul D. Hanna, Jul 29 2011: (Start)
O.g.f.: 1 - x + 3*x^2 - 27*x^3 + 441*x^4 - 11529*x^5 + 442827*x^6 -+ ... + a(n)*x^n + ...
O.g.f.: 1/(1 + x/(1 + 2*x/(1 + 9*x/(1 + 8*x/(1 + 25*x/(1 + 18*x/(1 + 49*x/(1 + 32*x/(1-...))))))))) (continued fraction). (End)

Mathematica

a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ JacobiCN[ x, 1/2], {x, 0, m}]]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, With[ {s = InverseSeries[ Integrate[ Series[(1 - x^4 / 4) ^ (-1/2), {x, 0, m + 1}], x]]}, m! SeriesCoefficient[ Sqrt[ (2 - s^2) / (2 + s^2)], {x, 0, m}]]]]; (* Michael Somos, Apr 25 2011 *)

Prog

(PARI) {a(n)=local(S=x, C); for(i=0, 2*n, S=intformal((1-2*S^2+O(x^(2*n+2)))^(3/4))); C=(1-2*S^2)^(1/4) ; (2*n)!*polcoeff(C, 2*n)}
(PARI) {a(n) = my(A, m); if( n<0, 0, m = 2*n; A = serreverse( intformal( (1 - x^4 / 4 + x * O(x^m)) ^ (-1/2))); m! * polcoeff( sqrt( (2 - A^2) / (2 + A^2)), m))}; /* Michael Somos, Apr 25 2011 */
(PARI) {a(n) = local(C=1+x); for(i=1, n, C = exp( intformal( C * intformal(-1/C^3 + x*O(x^n)) ) ) ); n!*polcoeff(C, n)}
for(n=0, 20, print1(a(2*n), ", "))
(PARI) {a(n) = local(C=1+x); for(i=1, n, C = exp( intformal( -1/C * intformal(C^3 + x*O(x^n)) ) ) ); n!*polcoeff(C, n)}
for(n=0, 20, print1(a(2*n), ", "))

Crossrefs

Cf. A159601 (S(x)); A193541, A193544: All of these have the same |a(n)|. - M. F. Hasler, Aug 31 2012

Cf. A129194.

Sequence in context: A108525 A136719 A279844 * A159601 A193541 A193544

Adjacent sequences: A159597 A159598 A159599 * A159601 A159602 A159603

Keyword

sign

Author

Paul D. Hanna, May 07 2009

Status

approved