A026378 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).

1, 4, 17, 75, 339, 1558, 7247, 34016, 160795, 764388, 3650571, 17501619, 84179877, 406020930, 1963073865, 9511333155, 46169418195, 224484046660, 1093097083475, 5329784874185, 26018549129545, 127154354598330, 622031993807565

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

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

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).

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

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

Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007

5th binomial transform of (-1)^n*A000108. - Paul Barry, Jan 13 2009

From Gary W. Adamson, May 17 2009: (Start)

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

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

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

From Tom Copeland, Nov 09 2014: (Start)

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).

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).

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Shu-Chiuan Chang, Robert Shrock, Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips, J. Stat. Physics 137 (2009) 667, table 5.
• D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
• Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
• Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
• J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
• Toufik Mansour, Jose Luis Ramirez, Enumration of Fuss-skew paths, Ann. Math. Inform. 55 (2022) 125-136, table 2, l=1.
• László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Crossrefs

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

Cf. A026375, A026376, A007317, A002212. - Gary W. Adamson, May 17 2009

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

Programs

Maple

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

Mathematica

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 *)
Table[Hypergeometric2F1[3/2, 1-n, 2, -4], {n, 1, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)

Formula

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

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

a(n) = coefficient of t^n in (1+t)(1+3t+t^2)^(n-1). - Emeric Deutsch, Jan 30 2004

a(n) = A026380(2n-2). - Emeric Deutsch, Feb 18 2004

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

a(n+1) = sum(k=0, n, binomial(n, k)*sum(i=0, k, binomial(k+i, i))). - Benoit Cloitre, Aug 06 2004

a(n+1) = sum(k=0, n, binomial(n, k)*binomial(2*k+1, k+1)). - Benoit Cloitre, Aug 06 2004

a(n) = Sum(k*A126182(n-1,k-1),k=1..n). - Emeric Deutsch, Jul 25 2007

From Paul Barry, Jan 13 2009: (Start)

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

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

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

a(n) ~ 5^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 08 2012

a(n) = hypergeom([3/2, 1-n], [2], -4). - Vladimir Reshetnikov, Apr 25 2016

a(n) = (-1)^n*(GegenbauerC(n-2,-n+1,3/2) - GegenbauerC(n-1,-n+1,3/2)). - Peter Luschny, May 13 2016