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%I #23 Mar 20 2014 16:04:28
%S 1,1,2,5,13,35,97,276,805,2404,7343,22916,72980,236857,782275,2625265,
%T 8938718,30834165,107608097,379454447,1350434278,4845475311,
%U 17512579630,63703732426,233063976059,857067469749,3166309373615,11745982220846
%N Number of Dyck paths of semilength n with no peak at height 4.
%D Peart and Woan, in press, G_4(x).
%H Vincenzo Librandi, <a href="/A059027/b059027.txt">Table of n, a(n) for n = 0..1000</a>
%H P. Peart and W.-J. Woan, <a href="https://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.: (2-3*x+x*(1-4*x)^(1/2))/(2-5*x+x*(1-4*x)^(1/2)).
%F a(n) = sum(k=1..n-2, sum(j=max(2*k-n+1,0)..k-1, (binomial(k,j)*((k-j)*binomial(2*n-3*k+j-3,n-1-2*k+j)))/(n-k-1)*2^j))+2^(n-1). - _Vladimir Kruchinin_, Oct 03 2013
%F a(n) ~ 4^n/(9*sqrt(Pi)*n^(3/2)) * (1+197/(24*n)). - _Vaclav Kotesovec_, Mar 20 2014
%e 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + ...
%t CoefficientList[Series[(2 - 3 x + x (1 - 4 x)^(1/2))/(2 - 5 x + x (1 - 4 x)^(1/2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 05 2013 *)
%o a(n):=if n=0 then 1 else sum(sum((binomial(k,j)*((k-j)*binomial(2*n-3*k+j-3,n-1-2*k+j)))/(n-k-1)*2^j,j,max(2*k-n+1,0),k-1),k,1,n-2)+2^(n-1); [_Vladimir Kruchinin_, Oct 03 2013]
%o (PARI) x='x+O('x^66); Vec((2-3*x+x*(1-4*x)^(1/2))/(2-5*x+x*(1-4*x)^(1/2))) \\ _Joerg Arndt_, Oct 03 2013
%Y G_1, G_2, G_3, G_4 give A000957, A000108, A059019, A059027 resp.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Feb 12 2001
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