A078531 - OEIS

%I #78 Mar 24 2023 15:06:24

%S 1,2,10,64,462,3584,29172,245760,2124694,18743296,168043980,

%T 1526726656,14025209100,130056978432,1215785268840,11445014102016,

%U 108401560073190,1032295389593600,9877854438949980,94927710773575680

%N Coefficients of power series that satisfies A(x)^2 - 4*x*A(x)^3 = 1, A(0)=1.

%C Radius of convergence of g.f. A(x) is r = 1/(2*3^(3/2)) where A(r) = sqrt(3).

%C 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

%C 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

%C 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

%H Robert Israel, <a href="/A078531/b078531.txt">Table of n, a(n) for n = 0..900</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry3/barry132.html">On the Central Coefficients of Bell Matrices</a>, J. Int. Seq. 14 (2011) # 11.4.3, example 10.

%H Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, <a href="http://dx.doi.org/10.1016/j.laa.2015.03.015">A combinatorial equivalence relation for formal power series</a>, Linear Algebra and its Applications, Available online 30 March 2015.

%H P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of non-crossing configurations</a>, Discrete Math., 204, 203-229, 1999.

%H Loïc Foissy, <a href="http://arxiv.org/abs/1504.06056">Free quadri-algebras and dual quadri-algebras</a>, arXiv preprint, 2015.

%H I. M. Gessel, <a href="http://arxiv.org/abs/1403.7656">A short proof of the Deutsch-Sagan congruence for connected non crossing graphs</a>, arXiv preprint arXiv:1403.7656, 2014

%H W. Mlotkowski, K. A. Penson and K. Zyczkowski, <a href="http://arxiv.org/abs/1211.7259">Densities of the Raney distributions</a>, arXiv preprint arXiv:1211.7259, 2012. - From _N. J. A. Sloane_, Jan 03 2013

%H V. U. Pierce, <a href="http://ercolani60.wdfiles.com/local--files/unoriented-ribbon-graphs-working-group/cont-toda.pdf">Continuum limits of Toda lattices for map enumeration</a>, in Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, edited by Anton Dzhamay, Ken'ichi Maruno, Virgil U. Pierce; Contemporary Mathematics, Vol. 593, 2013.

%H Vincent Pilaud, <a href="https://arxiv.org/abs/2205.06686">Pebble trees</a>, arXiv:2205.06686 [math.CO], 2022.

%H M. R. Sepanski, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p32">On Divisibility of Convolutions of Central Binomial Coefficients</a>, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.

%F 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

%F G.f.: (12x)^(-1) + (6x)^(-1)*sin(arcsin(216x^2-1)/3). - _Emeric Deutsch_, Nov 30 2002

%F a(n) = 2^(2n)*binomial(3n/2-1/2, n)/(n+1). - _Emeric Deutsch_, Dec 10 2002

%F 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

%F Sequence with offset 1 is expansion of reversion of g.f. x*sqrt(1-4x). - _Ralf Stephan_, Mar 22 2004

%F G.f. satisfies: A(x) = 1 / sqrt(1 - 4*x*A(x)).

%F G.f. satisfies: A(x) = Sum_{n>=0} ((2*n)!/n!^2)*x^n*A(x)^n. - _Paul D. Hanna_, Mar 03 2011

%F Self-convolution yields A214377, where A214377(n) = 4^n*binomial(3/2*n,n)*2/(n+2). - _Paul D. Hanna_, Jul 14 2012

%F 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

%F REVERSION transform of A002420 (both offsets 1). - _Michael Somos_, Jun 18 2014

%F 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

%F a(n) ~ 2^(n-1/2) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 03 2014

%F G.f. satisfies: 1-2*x*A(x)*C(x*A(x)) = 1/A(x), where C is g.f. of A000108. - _Werner Schulte_, Aug 07 2015

%F G.f.: (sqrt(3)/2)*(sech(arccosh(-sqrt(108)*x)/3)). - _Vladimir Kruchinin_, Oct 11 2022

%e G.f. = 1 + 2*x + 10*x^2 + 64*x^3 + 462*x^4 + 3584*x^5 + 29172*x^6 + ...

%e 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.

%p S:= series(RootOf(Z^2 - 4*x*Z^3-1,Z,1), x, 101):

%p seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Aug 07 2015

%t 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_ *)

%t 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 *)

%o (PARI) {a(n) = if( n<0, 0, n++; polcoeff( serreverse( x * sqrt( 1 - 4*x + O(x^n))), n))}; /* _Michael Somos_, Feb 05 2004 */

%o (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 */

%o (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 */

%o (Maxima)

%o taylor(sqrt(3)/2*(sech(acosh(-sqrt(108)*x)/3)),x,0,10); /* _Vladimir Kruchinin_ Oct 12 2022 */

%Y Cf. A002420, A078532, A078533, A078534, A078535, A214377.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 28 2002