%I #77 Mar 31 2023 17:42:57

%S 1,4,17,75,339,1558,7247,34016,160795,764388,3650571,17501619,

%T 84179877,406020930,1963073865,9511333155,46169418195,224484046660,

%U 1093097083475,5329784874185,26018549129545,127154354598330,622031993807565

%N a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=1; also a(n) = T(2n-1,n-1).

%C Number of lattice paths from (0,0) to the line x=n-1 that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps). Example: a(3)=17 because we have UD, UU, 9 HH paths, 3 HU paths and 3 UH paths. - _Emeric Deutsch_, Jan 22 2004

%C Also a(n) = number of integer strings s(0), ..., s(n) counted by array U in A026386 that have s(n)=1; a(n) = U(2n-1, n-1).

%C The Hankel transform of [1,1,4,17,75,339,1558,...] is [1,3,8,21,55,144,377,...] (see A001906). - _Philippe Deléham_, Apr 13 2007

%C Number of peaks in all skew Dyck paths of semilength n. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=4 because in the 3 (=A002212(2)) skew Dyck paths (UD)(UD), U(UD)D and U(UD)L we have altogether 4 peaks (shown between parentheses). - _Emeric Deutsch_, Jul 25 2007

%C Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,...]. - _Philippe Deléham_, Oct 24 2007

%C 5th binomial transform of (-1)^n*A000108. - _Paul Barry_, Jan 13 2009

%C From _Gary W. Adamson_, May 17 2009: (Start)

%C Convolved with A007317, (1, 2, 5, 15, 51, ...) = A026376: (1, 6, 30, 144, ...)

%C Equals A026375, (1, 3, 11, 45, 195, ...) convolved with A002212 prefaced with

%C a 1: (1, 1, 3, 10, 36, 137, ...). (End)

%C From _Tom Copeland_, Nov 09 2014: (Start)

%C The array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interpolating o.g.f. [1-sqrt(1-4x/(1+(1-t)x))]/2 and inverse x(1-x)/[1+(t-1)x(1-x)]. See A091867 for more info on this family. Here the interpolation is t=-4 (mod signs in the results).

%C Let C(x) = [1 - sqrt(1-4x)]/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).

%C O.g.f: G(x) = [-1 + sqrt(1 + 4*x/(1-5x))]/2 = -C[P(-x,5)].

%C Inverse O.g.f: Ginv(x) = x*(1+x)/[1 + 5x*(1+x)] = -P(Cinv(-x),-5) (signed A039717). (End)

%H Vincenzo Librandi, <a href="/A026378/b026378.txt">Table of n, a(n) for n = 1..1000</a>

%H Shu-Chiuan Chang, Robert Shrock, <a href="https://doi.org/10.1007/s10955-009-9868-0">Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips</a>, J. Stat. Physics 137 (2009) 667, table 5.

%H D. E. Davenport, L. W. Shapiro and L. C. Woodson, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i2p33">The Double Riordan Group</a>, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012

%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.

%H E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693">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 J. W. Layman, <a href="http://www.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 Toufik Mansour, Jose Luis Ramirez, <a href="https://doi.org/10.33039/ami.2022.01.002">Enumration of Fuss-skew paths</a>, Ann. Math. Inform. 55 (2022) 125-136, table 2, l=1.

%H László Németh, <a href="https://arxiv.org/abs/1905.13475">Tetrahedron trinomial coefficient transform</a>, arXiv:1905.13475 [math.CO], 2019.

%F G.f.: (1/2)/(5*x^2-x)*(1-5*x-(1-6*x+5*x^2)^(1/2)). E.g.f.: exp(3*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - _Vladeta Jovovic_, Oct 03 2003

%F G.f.: [(1-z)/sqrt(1-6z+5z^2)-1]/2 = z + 4z^2 + 17z^3 + ... - _Emeric Deutsch_, Jan 22 2004

%F a(n) = coefficient of t^n in (1+t)(1+3t+t^2)^(n-1). - _Emeric Deutsch_, Jan 30 2004

%F a(n) = A026380(2n-2). - _Emeric Deutsch_, Feb 18 2004

%F a(n) = [2(3n-2)a(n-1) - 5(n-2)a(n-2)]/n for n>=2; a(0)=0, a(1)=1. - _Emeric Deutsch_, Mar 18 2004

%F a(n+1) = sum(k=0, n, binomial(n, k)*sum(i=0, k, binomial(k+i, i))). - _Benoit Cloitre_, Aug 06 2004

%F a(n+1) = sum(k=0, n, binomial(n, k)*binomial(2*k+1, k+1)). - _Benoit Cloitre_, Aug 06 2004

%F a(n) = Sum(k*A126182(n-1,k-1),k=1..n). - _Emeric Deutsch_, Jul 25 2007

%F From _Paul Barry_, Jan 13 2009: (Start)

%F G.f.: (1/(1-5x))*c(-x/(1-5x)), c(x) the g.f. of A000108;

%F a(n) = sum{k=0..n, C(n,k)*(-1)^k*A000108(k)*5^(n-k)} (offset 0). (End)

%F G.f. 1/(1 - 3x - x(1 - x)/(1 - x - x(1 - x)/(1 - x - x(1 - x)/(1 - x - x(1 - x)/(1...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Jul 02 2010

%F a(n) ~ 5^(n-1/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 08 2012

%F a(n) = hypergeom([3/2, 1-n], [2], -4). - _Vladimir Reshetnikov_, Apr 25 2016

%F a(n) = (-1)^n*(GegenbauerC(n-2,-n+1,3/2) - GegenbauerC(n-1,-n+1,3/2)). - _Peter Luschny_, May 13 2016

%p a := n -> (-1)^n*simplify(GegenbauerC(n-2,-n+1,3/2) - GegenbauerC(n-1,-n+1,3/2)): seq(a(n), n=1..23); # _Peter Luschny_, May 13 2016

%t CoefficientList[Series[(1/2)/(5*x^2-x)*(1-5*x-(1-6*x+5*x^2)^(1/2)),{x,0,30}],x] (* _Vincenzo Librandi_, May 13 2012 *)

%t Table[Hypergeometric2F1[3/2, 1-n, 2, -4], {n, 1, 20}] (* _Vladimir Reshetnikov_, Apr 25 2016 *)

%Y Half the values of A026387. Bisection of A026380 and A026392.

%Y Cf. A026375, A026376, A007317, A002212. - _Gary W. Adamson_, May 17 2009

%Y Cf. A000108, A005043, A026375, A026380, A039717, A091867, A126182.

%K nonn

%O 1,2

%A _Clark Kimberling_