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Search: id:A159600
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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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| 1, -1, 3, -27, 441, -11529, 442827, -23444883, 1636819569, -145703137041, 16106380394643, -2164638920874507, 347592265948756521, -65724760945840254489, 14454276753061349098587
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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
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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).
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EXAMPLE
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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! +-...
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PROGRAM
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(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)}
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CROSSREFS
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Cf. A159601 (S(x)).
Sequence in context: A094577 A108525 A136719 this_sequence A159601 A111844 A118714
Adjacent sequences: A159597 A159598 A159599 this_sequence A159601 A159602 A159603
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KEYWORD
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sign
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), May 07 2009
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