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A078531
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Coefficients of power series that satisfies A(x)^2 - 4*x*A(x)^3 = 1, A(0)=1.
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6
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1, 2, 10, 64, 462, 3584, 29172, 245760, 2124694, 18743296, 168043980, 1526726656, 14025209100, 130056978432, 1215785268840, 11445014102016, 108401560073190, 1032295389593600, 9877854438949980, 94927710773575680
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OFFSET
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0,2
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COMMENTS
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Radius of convergence of g.f. A(x) is r = 1/(2*3^(3/2)) where A(r) = sqrt(3).
If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002
A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - Emeric Deutsch, Dec 10 2002
Number of symmetric non-crossing connected graphs on 2n+1 equidistant nodes on a circle (it is assumed that the axis of symmetry is a diameter of the circle passing through a given node). Example: a(1)=2 because on the nodes A,B,C (axis of symmetry through A) the only symmetric non-crossing connected graphs are {AB,AC} and {AB,AC,BC}. - Emeric Deutsch, Dec 03 2003
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REFERENCES
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P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.
W. Mlotkowski, K. A. Penson and K. Zyczkowski, Densities of the Raney distributions, arXiv preprint arXiv:1211.7259, 2012, - From N. J. A. Sloane, Jan 03 2013
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LINKS
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Table of n, a(n) for n=0..19.
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FORMULA
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a(n)=2*sum(binomial(3n-3, i)*binomial(2n-2-i, n), i=0..n-2)/(n-1) for n>1. - Emeric Deutsch, Nov 29 2002
G.f.: (12x)^(-1) + (6x)^(-1)*sin(arcsin(216x^2-1)/3). - Emeric Deutsch, Nov 30 2002
a(n)=2^(2n)*binomial(3n/2-1/2, n)/(n+1) - Emeric Deutsch, Dec 10 2002
G.f. A(x) = y satisfies y' * (6*x*y - 1) + 2*y^2 = 0, y' * (y^2 - 3) + 4*y^4 = 0. - Michael Somos, Feb 05 2004
Sequence with offset 1 is expansion of reversion of g.f. x*sqrt(1-4x). - R. Stephan, Mar 22 2004
G.f. satisfies: A(x) = Sum_{n>=0} ((2n)!/n!^2)*x^n*A(x)^n. [Paul D. Hanna, Mar 3 2011]
Self-convolution yields A214377, where A214377(n) = 4^n*binomial(3/2*n,n)*2/(n+2). - Paul D. Hanna, Jul 14 2012
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EXAMPLE
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1 + 2*x + 10*x^2 + 64*x^3 + 462*x^4 + 3584*x^5 + 29172*x^6 + 245760*x^7 + ...
A(x)^2 - 4x*A(x)^3 = 1 since A(x)^2 = 1 + 4x + 24x^2 + 148x^3 + 1280x^4 + 10296x^5 + ... and A(x)^3 = 1 + 6x + 42x^2 + 320x^3 + 2574x^4 + ... also a(1)=2^1, a(3)=2^6.
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MATHEMATICA
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a[n_] := 2^(2n)*Binomial[3n/2-1/2, n]/(n+1); Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jan 21 2013, after Emeric Deutsch *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n++; polcoeff( serreverse( x * sqrt( 1 - 4*x + O(x^n))), n))} /* Michael Somos, Feb 05 2004 */
(PARI) {a(n) = if( n<1, n==0, polcoeff( serreverse( x * (2 + x) / (4 * (1 + x)^3) + x * O(x^n)), n))} /* Michael Somos, Feb 05 2004 */
(PARI) {a(n)=local(B=sum(m=0, n, binomial(2*m, m)*x^m+x*O(x^n))); polcoeff(1/x*serreverse(x/B), n)} [Paul D. Hanna, Mar 3 2011]
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CROSSREFS
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Cf. A078532, A078533, A078534, A078535, A214377.
Sequence in context: A183165 A129130 A186268 * A223127 A130721 A167449
Adjacent sequences: A078528 A078529 A078530 * A078532 A078533 A078534
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Nov 28 2002
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STATUS
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approved
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