A129164 Sum of pyramid weights in all skew Dyck paths of semilength n.

1, 5, 22, 97, 436, 1994, 9241, 43257, 204052, 968440, 4619011, 22120630, 106300507, 512321437, 2475395302, 11986728457, 58156146652, 282640193312, 1375737276787, 6705522150972, 32724071280517, 159878425878847, 781910419686412, 3827639591654862, 18753350784435811

Offset: 1

Author

Emeric Deutsch, Apr 03 2007

Keywords

• nonn

Comments

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. A pyramid in a skew Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a skew Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a skew Dyck path (word) is the sum of the heights of its maximal pyramids.

Examples

			a(2)=5 because the pyramid weights of the paths (UD)(UD), (UUDD) and U(UD)L are 2, 2 and 1, respectively (the maximal pyramids are shown between parentheses).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Crossrefs

Cf. A026378, A129163.

Programs

Maple

G:=(1/sqrt(1-6*z+5*z^2)-1/(1-z))/2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..26);

Mathematica

Rest[CoefficientList[Series[(1/Sqrt[1-6*x+5*x^2]-1/(1-x))/2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := (Hypergeometric2F1[1/2, -n, 1, -4]-1)/2; Array[a, 25] (* Jean-François Alcover, Oct 11 2016, after Vladimir Kruchinin *)

Maxima

a(n):=n*sum(sum((binomial(-m+2*k-1,k-1)*binomial(n-1,k-1))/k,k,m,n), m,1,n); /* Vladimir Kruchinin, Oct 07 2011 */

Maxima

a(n):=sum(binomial(n,k)*binomial(2*k,k),k,1,n)/2; /* Vladimir Kruchinin, Oct 11 2016 */

Sage

A129164 = lambda n: n*hypergeometric([1, 3/2, 1-n], [2, 2], -4)
[simplify(A129164(n)) for n in (1..25)] # Peter Luschny, Sep 16 2014

Formula

a(n) = Sum_{k=1..n} k*A129163(n,k).

Partial sums of A026378.

G.f. = [1/sqrt(1-6*z+5*z^2)-1/(1-z)]/2.

a(n) = n*Sum_(m=1..n, Sum_(k=m..n,(binomial(-m+2*k-1,k-1)*binomial(n-1,k-1))/k)). - Vladimir Kruchinin, Oct 07 2011

Recurrence: n*a(n) = (7*n-4)*a(n-1) - (11*n-14)*a(n-2) + 5*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 5^(n+1/2)/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012

a(n) = n*hypergeometric([1, 3/2, 1-n], [2, 2], -4). - Peter Luschny, Sep 16 2014

a(n) = Sum_{k=1..n} (binomial(n,k)*binomial(2*k,k))/2. - Vladimir Kruchinin, Oct 11 2016