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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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14
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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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LINKS
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Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.
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FORMULA
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a(n) = 2*(sum_{i=0..n-2} binomial(3n-3, i)*binomial(2n-2-i, n))/(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). - Ralf Stephan, Mar 22 2004
G.f. satisfies: A(x) = 1 / sqrt(1 - 4*x*A(x)).
G.f. satisfies: A(x) = Sum_{n>=0} ((2*n)!/n!^2)*x^n*A(x)^n. - Paul D. Hanna, Mar 03 2011
D-finite with recurrence n*(n+1)*a(n) + n*(n-1)*a(n-1) - 12*(3*n-1)*(3*n-5)*a(n-2) - 12*(3*n-4)*(3*n-8)*a(n-3) = 0. - R. J. Mathar, Jun 07 2013
0 = a(n)*(16*a(n+1) - 10*a(n+2)) + a(n+1)*(2*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Jun 18 2014
a(n) ~ 2^(n-1/2) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 03 2014
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EXAMPLE
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G.f. = 1 + 2*x + 10*x^2 + 64*x^3 + 462*x^4 + 3584*x^5 + 29172*x^6 + ...
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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MAPLE
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S:= series(RootOf(Z^2 - 4*x*Z^3-1, Z, 1), x, 101):
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MATHEMATICA
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a[ n_] := With[ {m = n + 1}, If[ m < 1, 0, SeriesCoefficient[ InverseSeries @ Series[ x Sqrt[1 - 4 x], {x, 0, m}], {x, 0, m}]]]; (* Michael Somos, Jun 18 2014 *)
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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 03 2011 */
(Maxima)
taylor(sqrt(3)/2*(sech(acosh(-sqrt(108)*x)/3)), x, 0, 10); /* Vladimir Kruchinin Oct 12 2022 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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