login
A129164
Sum of pyramid weights in all skew Dyck paths of semilength n.
3
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
1,2
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
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
Sequence in context: A297333 A129158 A342554 * A123347 A087439 A033452
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 03 2007
STATUS
approved