1,2

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 (deutsch(AT)duke.poly.edu), 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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 (deutsch(AT)duke.poly.edu), Jul 25 2007

Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007

5th binomial transform of (-1)^n*A000108. [From Paul Barry (pbarry(AT)wit.ie), Jan 13 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), 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)

E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

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 (vladeta(AT)eunet.rs), Oct 03 2003

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

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

a(n)=A026380(2n-2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 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 (deutsch(AT)duke.poly.edu), Mar 18 2004

a(n+1) = sum(k=0, n, binomial(n, k)*sum(i=0, k, binomial(k+i, i))) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2004

a(n+1) = sum(k=0, n, binomial(n, k)*binomial(2*k+1, k+1)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2004

a(n)=Sum(k*A126182(n-1,k-1),k=1..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 25 2007

Contribution from Paul Barry (pbarry(AT)wit.ie), 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)

Cf. A002212, A026375.

Cf. A026380.

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

Cf. A126182.

A026375, A026376, A007317, A002212 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]

Sequence in context: A049027 A026751 A081568 this_sequence A151247 A117439 A081910

Adjacent sequences: A026375 A026376 A026377 this_sequence A026379 A026380 A026381

nonn

Clark Kimberling (ck6(AT)evansville.edu)