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A059027
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Number of Dyck paths of semilength n with no peak at height 4.
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2
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1, 1, 2, 5, 13, 35, 97, 276, 805, 2404, 7343, 22916, 72980, 236857, 782275, 2625265, 8938718, 30834165, 107608097, 379454447, 1350434278, 4845475311, 17512579630, 63703732426, 233063976059, 857067469749, 3166309373615, 11745982220846
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OFFSET
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0,3
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REFERENCES
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Peart and Woan, in press, G_4(x).
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LINKS
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FORMULA
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G.f.: (2-3*x+x*(1-4*x)^(1/2))/(2-5*x+x*(1-4*x)^(1/2)).
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
a(n) ~ 4^n/(9*sqrt(Pi)*n^(3/2)) * (1+197/(24*n)). - Vaclav Kotesovec, Mar 20 2014
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EXAMPLE
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1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + ...
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MATHEMATICA
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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 *)
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PROG
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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]
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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