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%I A026671 #92 Sep 15 2022 20:55:02
%S A026671 1,3,11,43,173,707,2917,12111,50503,211263,885831,3720995,15652239,
%T A026671 65913927,277822147,1171853635,4945846997,20884526283,88224662549,
%U A026671 372827899079,1576001732485,6663706588179,28181895551161,119208323665543,504329070986033,2133944799315027
%N A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).
%C A026671 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 A026671 From _Paul Barry_, Jan 25 2009: (Start)
%C A026671 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 A026671 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 A026671 Binomial transform of A111961. - _Philippe Deléham_, Feb 11 2009
%C A026671 From _Paul Barry_, Nov 03 2010: (Start)
%C A026671 The sequence 1,1,3,... has g.f. 1/(1-x/sqrt(1-4x)), INVERT transform of A000984.
%C A026671 It is an eigensequence of the sequence array for A000984. (End)
%D A026671 L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
%H A026671 Vincenzo Librandi, <a href="/A026671/b026671.txt">Table of n, a(n) for n = 0..200</a>
%H A026671 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 A026671 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 A026671 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 A026671 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 A026671 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 A026671 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 A026671 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 A026671 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 A026671 From _Wolfdieter Lang_, Mar 21 2000: (Start)
%F A026671 G.f.: 1/(sqrt(1-4*x)-x).
%F A026671 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 A026671 G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. - _Michael Somos_, Apr 20 2007
%F A026671 From _Paul Barry_, Jan 25 2009: (Start)
%F A026671 G.f.: 1/(1 - 3xc(x) + x^2*c(x)^2);
%F A026671 G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction).
%F A026671 a(0) = 1, a(n) = Sum_{k=0..n} (k/(2n-k))*C(2n-k,n-k)*F(2k+2). (End)
%F A026671 a(n) = Sum_{k=0..n} A039599(n,k)*A000045(k+2). - _Philippe Deléham_, Feb 11 2009
%F A026671 From _Paul Barry_, Feb 08 2009: (Start)
%F A026671 G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction);
%F A026671 G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
%F A026671 From _Gary W. Adamson_, Jul 14 2011: (Start)
%F A026671 a(n) = the upper left term in M^n, M = the infinite square production matrix:
%F A026671   3, 2, 0, 0, 0, 0, ...
%F A026671   1, 1, 1, 0, 0, 0, ...
%F A026671   1, 1, 1, 1, 0, 0, ...
%F A026671   1, 1, 1, 1, 1, 0, ...
%F A026671   1, 1, 1, 1, 1, 1, ...
%F A026671   ...
%F A026671 (End)
%F A026671 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 A026671 a(n) ~ (2+sqrt(5))^n/sqrt(5). - _Vaclav Kotesovec_, Oct 08 2012
%t A026671 Table[SeriesCoefficient[1/(Sqrt[1-4*x]-x),{x,0,n}],{n,0,30}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%o A026671 (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 A026671 (PARI) x='x+O('x^66); Vec( 1/(sqrt(1-4*x)-x) ) \\ _Joerg Arndt_, May 04 2013
%o A026671 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(Sqrt(1-4*x)-x) )); // _G. C. Greubel_, Jul 16 2019
%o A026671 (Sage) (1/(sqrt(1-4*x)-x)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 16 2019
%o A026671 (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 A026671 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 A026671 Cf. A026781.
%K A026671 nonn,easy
%O A026671 0,2
%A A026671 _Clark Kimberling_, _Miklos Bona_

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