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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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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

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Last modified January 27 12:01 EST 2020. Contains 331295 sequences. (Running on oeis4.)