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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).
5
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
OFFSET
0,5
COMMENTS
The positive terms form the partial sums of A000344.
LINKS
FORMULA
a(n) = Sum_{k>=0} k*A097607(n,k).
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 */
(Python)
from functools import cache
@cache
def B(n, k):
if n <= 0 or k <= 0: return 0
if n == k: return 1
return B(n - 1, k) + B(n, k - 1)
def A143955(k):
return B(k + 3, k - 2)
print([A143955(n) for n in range(26)]) # Peter Luschny, May 15 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 14 2008
STATUS
approved