OFFSET
1,2
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.
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
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
EXAMPLE
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).
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 03 2007
STATUS
approved