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A159600
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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.
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5
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1, -1, 3, -27, 441, -11529, 442827, -23444883, 1636819569, -145703137041, 16106380394643, -2164638920874507, 347592265948756521, -65724760945840254489, 14454276753061349098587
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Coefficients in the expansion of the cosine lemniscate function. - Michael Somos, Apr 25 2011
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...
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FORMULA
| 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
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).
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].
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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! +-...
...
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). [From Paul D. Hanna, Jul 29 2011].
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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 *)
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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) = local(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 */
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CROSSREFS
| Cf. A159601 (S(x)); A193541, A193544.
Sequence in context: A108525 A136719 * A159601 A193541 A193544 A111844
Adjacent sequences: A159597 A159598 A159599 * A159601 A159602 A159603
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KEYWORD
| sign
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), May 07 2009
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