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%I M1627 #78 Dec 11 2022 14:25:27
%S 1,2,6,18,60,200,700,2450,8820,31752,116424,426888,1585584,5889312,
%T 22084920,82818450,312869700,1181952200,4491418360,17067389768,
%U 65166397296,248817153312,953799087696,3656229836168,14062422446800,54086240180000,208618354980000
%N Number of walks of length n on square lattice, starting at origin, staying in first quadrant.
%C a(n) is the number of involutions of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 5. - _Eric S. Egge_, Oct 21 2008
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A. Bostan, <a href="http://www-apr.lip6.fr/sem-comb-slides/IHP-bostan.pdf">Computer Algebra for Lattice Path Combinatorics</a>, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.
%H Alin Bostan, <a href="https://specfun.inria.fr/bostan/HDR.pdf">Calcul Formel pour la Combinatoire des Marches</a> [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017.
%H Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien <a href="https://doi.org/10.1016/j.ejc.2016.10.010">Hypergeometric expressions for generating functions of walks with small steps in the quarter plane.</a> Eur. J. Comb. 61, 242-275 (2017)
%H R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>
%H R. K. Guy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%F a(n) = binomial(n, floor(n/2))*binomial(n+1, floor((n+1)/2)).
%F E.g.f.: (BesselI(0, 2*x)+BesselI(1, 2*x))^2. - _Vladeta Jovovic_, Apr 28 2003
%F EXPCONV of A001405 with itself, i.e., a(n) = sum_{k=0}^n binomial(n,k)*A001405(k)*A001405(n-k). - _Max Alekseyev_, May 18 2006
%F G.f.: (16*x^2-1)*hypergeom([3/2, 3/2],[2],16*x^2) + (1/(2x)+2)*hypergeom([1/2, 1/2],[1],16*x^2) - 1/(2x). - _Mark van Hoeij_, Oct 13 2009
%F G.f.: (hypergeom([1/2,1/2],[1],16*x^2) - 1)/(2*x) + hypergeom([1/2,3/2],[2],16*x^2). - _Mark van Hoeij_, Aug 14 2014
%F a(n) = A241530(n)*2*floor(n/2)/(floor(n/2)+1). - _Peter Luschny_, Apr 25 2014
%F D-finite with recurrence (n+2)*(n+1)*a(n) +4*(-2*n-1)*a(n-1) -16*n*(n-1)*a(n-2)=0. - _R. J. Mathar_, Mar 07 2015
%F 0 = a(n)*(+16*a(n+2) -6*a(n+3)) +a(n+1)*(-2*a(n+2) +a(n+3)) if n >= 0. - _Michael Somos_, Oct 17 2019
%e G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 60*x^4 + 200*x^5 + 700*x^6 + 2450*x^7 + ... - _Michael Somos_, Oct 17 2019
%t f[n_] := Binomial[n, Floor[n/2]] Binomial[n + 1, Floor[(n + 1)/2]]; Array[f, 25, 0] (* _Robert G. Wilson v_ *)
%o (Magma) [Binomial(n, Floor(n/2))*Binomial(n+1, Floor((n+1)/2)): n in [0..30]]; // _Vincenzo Librandi_, Feb 18 2015
%Y Cf. A001700, A060897-A060900.
%Y a(2*n) = A000894(n), a(2*n+1) = 2*A060150(n+1).
%K nonn,walk
%O 0,2
%A _N. J. A. Sloane_
%E Additional comments from _David W. Wilson_, May 05 2001
%E a(25)-a(26) from _Vincenzo Librandi_, Feb 18 2015
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