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%I #54 Jul 25 2020 02:31:08
%S 1,1,2,4,9,22,58,163,483,1494,4783,15740,52956,181391,630533,2218761,
%T 7888266,28291588,102237141,371884771,1360527143,5002837936,
%U 18479695171,68539149518,255137783916,952914971191,3569834343547,13410481705795
%N Number of Dyck paths of semilength n with no peak at height 3.
%C From _Gary W. Adamson_, Jul 11 2011: (Start)
%C a(n) is the upper left term in M^n, where M is an infinite square production matrix in which a column of (1,1,0,0,0,...) is prepended to an infinite lower triangular matrix of all 1's and the rest zeros, as follows:
%C 1, 1, 0, 0, 0, 0, ...
%C 1, 1, 1, 0, 0, 0, ...
%C 0, 1, 1, 1, 0, 0, ...
%C 0, 1, 1, 1, 1, 0, ...
%C 0, 1, 1, 1, 1, 1, ...
%C ... (End)
%H G. C. Greubel, <a href="/A059019/b059019.txt">Table of n, a(n) for n = 0..1000</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/PEART/pwatjis2.html">Dyck Paths With No Peaks at Height k</a>, J. Integer Sequences, 4 (2001), #01.1.3.
%F G.f.: 2/(2 - 3*x + x*(1-4*x)^(1/2)).
%F 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
%F a(n) ~ 2^(2*n + 2)/(25*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Dec 10 2013
%F a(n+1) - a(n) = A135337(n), n > 0. - _Philippe Deléham_, Oct 02 2014
%e 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 58*x^6 + ...
%p 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
%t CoefficientList[Series[2/(2-3*x+x*Sqrt[1-4*x]), {x, 0, 20}], x]
%o (Maxima)
%o 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
%o (PARI) x='x+O('x^66); Vec( 2/(2-3*x+x*(1-4*x)^(1/2)) ) \\ _Joerg Arndt_, Oct 02 2013
%Y G_1, G_2, G_3, G_4 give A000957, A000108, A059019, A059027, respectively.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Feb 12 2001
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