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A105864
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Expansion of (1/(1-x^2))*c(x/(1-x^2)), where c(x) is the g.f. of A000108.
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2
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1, 1, 3, 7, 21, 65, 215, 735, 2585, 9281, 33883, 125383, 469229, 1772801, 6752623, 25902975, 99978865, 388001025, 1513077235, 5926139207, 23301146501, 91942524481, 363957103303, 1444966207967, 5752187960841, 22955311342145
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OFFSET
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0,3
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COMMENTS
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The radius of convergence r of the g.f. A(x) satisfies: r = (1-r^2)/4 = lim_{n->inf} a(n)/a(n+1) = sqrt(5) - 2 = 0.2360679... with A(r) = 1/(2*r) = (sqrt(5) + 2)/2 = 2.1180339... - Paul D. Hanna, Sep 06 2011
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LINKS
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FORMULA
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G.f.: (1 - sqrt((1 - 4*x - x^2)/(1 - x^2)))/(2*x).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k) * C(n-2*k)}.
G.f. satisfies: A(x) = 1/(1-x^2) + x*A(x)^2. - Paul D. Hanna, Sep 06 2011
Conjecture: (n+1)*a(n) + 2*(1-2*n)*a(n-1) + 2*(1-n)*a(n-2) + 2*(2*n-3)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011
G.f.: (1-1/G(0))/(2*x), where G(k) = 1 + 4*x*(4*k+1)/( (1-x^2)*(4*k+2) - x*(1-x^2)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (1-x^2)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) ~ 5^(1/4)*(2+sqrt(5))^(n+1)/(4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 16 2013
G.f.: 1/G(x), where G(x) = 1 - x^2 - (x - x^3)/ G(x) (continued fraction). - Nikolaos Pantelidis, Jan 08 2023
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MATHEMATICA
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a[0] = a[1] = 1; a[2] = 3; a[3] = 7; a[n_] := a[n] = (-((n-3)*a[n-4]) - 2*(2*n-3)*a[n-3] + 2*(n-1)*a[n-2] + 2*(2*n-1)*a[n-1])/(n+1); Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 09 2017, using "FindSequenceFunction" *)
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PROG
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(PARI) {a(n)=polcoeff((1-sqrt(1-4*x/(1-x^2 +O(x^(n+2)))))/(2*x), n)} /* Paul D. Hanna */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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