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A143955 Sum of the altitudes of the leftmost valleys of all Dyck paths of semilength n (if path has no valley, then this altitude is taken to be 0). 3
0, 0, 0, 1, 6, 26, 101, 376, 1377, 5017, 18277, 66727, 244377, 898129, 3312554, 12260129, 45526754, 169588754, 633580634, 2373550184, 8914719134, 33562602134, 126640791884, 478848661898, 1814142235028, 6885560250148 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n)=Sum(k*A097607(n,k), k>=0).

The positive terms form the partial sums of A000344.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

G.f.: z^3*C^5/(1-z), where C=(1-sqrt(1-4*z))/(2*z) is the generating function of the Catalan numbers (A000108).

Conjecture: (n+2)*a(n) -4*(2*n+1)*a(n-1) +2*(10*n-9)*a(n-2) +17*(2-n)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 24 2012

a(n) ~ 5*4^n/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014

a(n) = 5*Sum_{k=2..n-1}(binomial(2*k,k-2)/(k+3)). - Vladimir Kruchinin, Mar 15 2016

EXAMPLE

a(4)=6 because the Dyck paths of semilength 4 with leftmost valley at a positive altitude are UUDUDDUD, UUDUDUDD, UUDUUDDD, UUUDDUDD and UUUDUDDD, where U=(1,1) and D=(1,-1); these altitudes are 1, 1, 1, 1 and 2, respectively.

MAPLE

C:=((1-sqrt(1-4*z))*1/2)/z: G:=z^3*C^5/(1-z): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..27);

MATHEMATICA

CoefficientList[Series[x^3 ((1 - (1 - 4 x)^(1/2))/(2 x))^5/(1 - x), {x, 0, 40}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

PROG

(Maxima)

a(n):=5*sum(binomial(2*k, k-2)/(k+3), k, 2, n-1); /* Vladimir Kruchinin, Mar 15 2016 */

CROSSREFS

Cf. A000108, A000344, A097607.

Sequence in context: A290347 A034560 A307306 * A256428 A196860 A254317

Adjacent sequences:  A143952 A143953 A143954 * A143956 A143957 A143958

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 14 2008

STATUS

approved

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Last modified April 1 10:03 EDT 2020. Contains 333159 sequences. (Running on oeis4.)