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Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|k>0,r>0} which never go above the line y=x.
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%I #41 Dec 08 2021 10:31:20

%S 1,2,10,63,454,3539,29008,246255,2145722,19078536,172402396,

%T 1578687082,14616730080,136606848093,1287022395324,12210382758519,

%U 116553763025178,1118580919711060,10786838228669692,104469304517331666,1015700422725526916

%N Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|k>0,r>0} which never go above the line y=x.

%H Vincenzo Librandi, <a href="/A175962/b175962.txt">Table of n, a(n) for n = 0..200</a>

%H Joseph P. S. Kung and Anna de Mier, <a href="http://arxiv.org/abs/1109.1806">Rook and queen paths with boundaries</a>, arXiv:1109.1806 [math.CO], 2011.

%H J. P. S. Kung and A. de Mier, <a href="http://dx.doi.org/10.1016/j.jcta.2012.08.010">Catalan lattice paths with rook, bishop and spider steps</a>, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From _N. J. A. Sloane_, Dec 27 2012

%F G.f.: ((1-t)*(1+t-4*t^2)-(1-t)^2*sqrt(1-12*t+16*t^2))/(2*t*(2 - 3*t)^2). [Kung-de Mier]. - corrected by _Vaclav Kotesovec_, Sep 07 2012

%F Apparently 2*n*(n+1)*a(n) -n*(29*n-10)*a(n-1) +19*n*(5*n-7)*a(n-2) -2*n*(58*n-149)*a(n-3) +48*n*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jul 24 2012

%F a(n) ~ 5/2*sqrt(246*sqrt(5)-550)/sqrt(Pi) * (6+2*sqrt(5))^n/n^(3/2). - _Vaclav Kotesovec_, Nov 01 2012

%F Equivalently, a(n) ~ 5^(5/4) * 2^(2*n) * phi^(2*n - 5) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 08 2021

%t Table[SeriesCoefficient[((1-t)*(1+t-4t^2)-(1-t)^2*Sqrt[1-12t+16t^2])/(2t*(2-3t)^2), {t,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 07 2012 *)

%o (PARI) x='x+O('x^50); Vec(((1-t)*(1+t-4*t^2)-(1-t)^2*sqrt(1-12*t+16*t^2))/(2*t*(2 - 3*t)^2)) \\ _G. C. Greubel_, Mar 22 2017

%Y Cf. A175912, A175939.

%K nonn

%O 0,2

%A _Eric Werley_, Dec 06 2010

%E Edited by _N. J. A. Sloane_, Sep 24 2011

%E Minor edits by _Vaclav Kotesovec_, Mar 31 2014