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%I #92 Sep 15 2022 20:55:02
%S 1,3,11,43,173,707,2917,12111,50503,211263,885831,3720995,15652239,
%T 65913927,277822147,1171853635,4945846997,20884526283,88224662549,
%U 372827899079,1576001732485,6663706588179,28181895551161,119208323665543,504329070986033,2133944799315027
%N Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).
%C 1, 1, 3, 11, 43, 173, ... is the unique sequence for which both the Hankel transform of the sequence itself and the Hankel transform of its left shift are the powers of 2 (A000079). For example, det[{{1, 1, 3}, {1, 3, 11}, {3, 11, 43}}] = det[{{1, 3, 11}, {3, 11, 43}, {11, 43, 173}}] = 4. - _David Callan_, Mar 30 2007
%C From _Paul Barry_, Jan 25 2009: (Start)
%C a(n) is the image of F(2n+2) under the Catalan matrix (1,xc(x)) where c(x) is the g.f. of A000108.
%C The sequence 1,1,3,... is the image of A001519 under (1,xc(x)). This sequence has g.f. given by 1/(1-x-2x^2/(1-3x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). (End)
%C Binomial transform of A111961. - _Philippe Deléham_, Feb 11 2009
%C From _Paul Barry_, Nov 03 2010: (Start)
%C The sequence 1,1,3,... has g.f. 1/(1-x/sqrt(1-4x)), INVERT transform of A000984.
%C It is an eigensequence of the sequence array for A000984. (End)
%D L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
%H Vincenzo Librandi, <a href="/A026671/b026671.txt">Table of n, a(n) for n = 0..200</a>
%H Jean-Christophe Aval, Adrien Boussicault and Sandrine Dasse-Hartaut, <a href="http://arxiv.org/abs/1109.4907">The tree structure in staircase tableaux</a>, arXiv:1109.4907 [math.CO], 2011-2013.
%H Cyril Banderier, Markus Kuba, and Michael Wallner, <a href="https://arxiv.org/abs/2103.03751">Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions</a>, arXiv:2103.03751 [math.PR], 2021.
%H Miklos Bona, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v5i1r31/0">The permutation classes equinumerous to the smooth class</a>, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.
%H David Callan and Toufik Mansour, <a href="http://arxiv.org/abs/1602.05182">Five subsets of permutations enumerated as weak sorting permutations</a>, arXiv:1602.05182 [math.CO], 2016.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H Huyile Liang, Jeffrey Remmel and Sainan Zheng, <a href="https://arxiv.org/abs/1710.05795">Stieltjes moment sequences of polynomials</a>, arXiv:1710.05795 [math.CO], 2017, see page 16.
%F From _Wolfdieter Lang_, Mar 21 2000: (Start)
%F G.f.: 1/(sqrt(1-4*x)-x).
%F a(n) = Sum_{i=1..n} a(i-1)*binomial(2*(n-i), n-i) + binomial(2*n, n), n >= 1, a(0)=1. (End)
%F G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. - _Michael Somos_, Apr 20 2007
%F From _Paul Barry_, Jan 25 2009: (Start)
%F G.f.: 1/(1 - 3xc(x) + x^2*c(x)^2);
%F G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction).
%F a(0) = 1, a(n) = Sum_{k=0..n} (k/(2n-k))*C(2n-k,n-k)*F(2k+2). (End)
%F a(n) = Sum_{k=0..n} A039599(n,k)*A000045(k+2). - _Philippe Deléham_, Feb 11 2009
%F From _Paul Barry_, Feb 08 2009: (Start)
%F G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction);
%F G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
%F From _Gary W. Adamson_, Jul 14 2011: (Start)
%F a(n) = the upper left term in M^n, M = the infinite square production matrix:
%F 3, 2, 0, 0, 0, 0, ...
%F 1, 1, 1, 0, 0, 0, ...
%F 1, 1, 1, 1, 0, 0, ...
%F 1, 1, 1, 1, 1, 0, ...
%F 1, 1, 1, 1, 1, 1, ...
%F ...
%F (End)
%F D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 3*(5*n-8)*a(n-2) - 2*(2*n-3)*a(n-3). - _Vaclav Kotesovec_, Oct 08 2012
%F a(n) ~ (2+sqrt(5))^n/sqrt(5). - _Vaclav Kotesovec_, Oct 08 2012
%t Table[SeriesCoefficient[1/(Sqrt[1-4*x]-x),{x,0,n}],{n,0,30}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%o (PARI) {a(n)= if(n<0, 0, polcoeff( 1/(sqrt(1 -4*x +x*O(x^n)) -x), n))} /* _Michael Somos_, Apr 20 2007 */
%o (PARI) x='x+O('x^66); Vec( 1/(sqrt(1-4*x)-x) ) \\ _Joerg Arndt_, May 04 2013
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(Sqrt(1-4*x)-x) )); // _G. C. Greubel_, Jul 16 2019
%o (Sage) (1/(sqrt(1-4*x)-x)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 16 2019
%o (GAP) a:=[3,11,43];; for n in [4..30] do a[n]:=(2*(4*n-3)*a[n-1] - 3*(5*n-8)*a[n-2] - 2*(2*n-3)*a[n-3])/n; od; Concatenation([1], a); # _G. C. Greubel_, Jul 16 2019
%Y a(n)=T(2n-1,n-1), T given by A026736, a(n)=T(2n,n), T given by A026670, a(n)=T(2n+1,n+1), T given by A026725. Row sums of triangle A054335.
%Y Cf. A026781.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, _Miklos Bona_
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