OFFSET
0,4
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
FORMULA
a(n) = Sum_{k=0..n-1} k*A100754(n,k).
G.f.: z^2*C/((1-z*C+z)^2*(1-2*z*C)), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) ~ 2^(2*n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
a(n) = Sum_{k=1..n} (k*(-1)^(k+1)*binomial(2*n-k,n-k-1)). - Vladimir Kruchinin, Oct 22 2016
D-finite with recurrence 2*(n+1)*a(n) -9*n*a(n-1) -3*n*a(n-2) +5*(5*n-16)*a(n-3) +6*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jul 22 2022
EXAMPLE
a(3)=3 because in the two hill-free Dyck paths of semilength 3, namely U(UD)(UD)D and UU(UD)DD, we have altogether 3 peaks (shown between parentheses).
MAPLE
C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z)^2*z^2*C/(1-2*z*C): Gser:=series(G, z=0, 32): 0, seq(coeff(Gser, z^n), n=1..28);
MATHEMATICA
CoefficientList[Series[1/(1-x*(1-Sqrt[1-4*x])/2/x+x)^2*x^2*(1-Sqrt[1-4*x])/2/x/(1-2*x*(1-Sqrt[1-4*x])/2/x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
a[n_] := If[n<=1, 0, Binomial[2n-1, n-2] Hypergeometric2F1[2, 2-n, 1-2n, -1]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 22 2016, after Vladimir Kruchinin *)
PROG
(Maxima)
a(n):=sum(k*(-1)^(k+1)*binomial(2*n-k, n-k-1), k, 1, n); /* Vladimir Kruchinin, Oct 22 2016 */
(PARI) my(x='x + O('x^50)); concat([0, 0], Vec((2*x*(1-sqrt(1-4*x)))/(sqrt(1-4*x)*(1 + 2*x + sqrt(1-4*x))^2))) \\ G. C. Greubel, Feb 08 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 04 2005
STATUS
approved