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A114515 Number of peaks in all hill-free Dyck paths of semilength n (a Dyck path is hill-free if it has no peaks at level 1). 3
0, 0, 1, 3, 12, 45, 171, 651, 2488, 9540, 36690, 141482, 546864, 2118207, 8219967, 31952115, 124389552, 484908408, 1892657934, 7395597354, 28928182440, 113260606074, 443827115886, 1740592240638, 6831289801872, 26829201570600 (list; graph; refs; listen; history; text; internal format)
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-zC+z)^2*(1-2zC)}, where C=[1-sqrt(1-4z)]/(2z) 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

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-Fran├žois 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

Cf. A100754.

Sequence in context: A128593 A085481 A030195 * A192467 A151162 A094547

Adjacent sequences:  A114512 A114513 A114514 * A114516 A114517 A114518

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 04 2005

STATUS

approved

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Last modified September 22 15:04 EDT 2021. Contains 347607 sequences. (Running on oeis4.)