

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
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OFFSET

1,2


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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, 155176.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203


FORMULA

a(n) = Sum_{k=1..n} k*A129163(n,k).
Partial sums of A026378.
G.f. = [1/sqrt(16*z+5*z^2)1/(1z)]/2.
a(n) = n*Sum_(m=1..n, Sum_(k=m..n,(binomial(m+2*k1,k1)*binomial(n1,k1))/k)).  Vladimir Kruchinin, Oct 07 2011
Recurrence: n*a(n) = (7*n4)*a(n1)  (11*n14)*a(n2) + 5*(n2)*a(n3).  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, 1n], [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(16*z+5*z^2)1/(1z))/2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..26);


MATHEMATICA

Rest[CoefficientList[Series[(1/Sqrt[16*x+5*x^2]1/(1x))/2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := (Hypergeometric2F1[1/2, n, 1, 4]1)/2; Array[a, 25] (* JeanFrançois Alcover, Oct 11 2016, after Vladimir Kruchinin *)


PROG

(Maxima)
a(n):=n*sum(sum((binomial(m+2*k1, k1)*binomial(n1, k1))/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, 1n], [2, 2], 4)
[simplify(A129164(n)) for n in (1..25)] # Peter Luschny, Sep 16 2014


CROSSREFS

Cf. A026378, A129163.
Sequence in context: A200676 A297333 A129158 * A123347 A087439 A033452
Adjacent sequences: A129161 A129162 A129163 * A129165 A129166 A129167


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Apr 03 2007


STATUS

approved



