A059019 Number of Dyck paths of semilength n with no peak at height 3.
1, 1, 2, 4, 9, 22, 58, 163, 483, 1494, 4783, 15740, 52956, 181391, 630533, 2218761, 7888266, 28291588, 102237141, 371884771, 1360527143, 5002837936, 18479695171, 68539149518, 255137783916, 952914971191, 3569834343547, 13410481705795
Offset: 0
Keywords
Examples
1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 58*x^6 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
- Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
- P. Peart and W.-J. Woan, Dyck Paths With No Peaks at Height k, J. Integer Sequences, 4 (2001), #01.1.3.
Programs
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Maple
A059019:=n->add(add(binomial(k,j)*j*binomial(2*n-2*k-j-1,n-k-j)/(n-k),j=0..min(k,n-k)), k=1..n-1)+1: seq(A059019(n),n=0..30); # Wesley Ivan Hurt, Oct 01 2014
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Mathematica
CoefficientList[Series[2/(2-3*x+x*Sqrt[1-4*x]), {x, 0, 20}], x] -
Maxima
a(n):=sum(sum(binomial(k,j)*j*binomial(2*n-2*k-j-1,n-k-j),j,0,min(k,n-k))/(n-k),k,1,n-1)+1; \\ Vladimir Kruchinin, Oct 02 2013
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PARI
x='x+O('x^66); Vec( 2/(2-3*x+x*(1-4*x)^(1/2)) ) \\ Joerg Arndt, Oct 02 2013
Formula
G.f.: 2/(2 - 3*x + x*(1-4*x)^(1/2)).
a(n) = Sum_{k=1..n-1} (Sum_{j=0..min(k,n-k)} binomial(k,j)*j*binomial(2*n-2*k-j-1, n-k-j) /(n-k)) + 1. - Vladimir Kruchinin, Oct 02 2013
a(n) ~ 2^(2*n + 2)/(25*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 10 2013
a(n+1) - a(n) = A135337(n), n > 0. - Philippe Deléham, Oct 02 2014
Comments