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A114515
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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).
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3
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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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) = Sum_{k=1..n} (k*(-1)^(k+1)*binomial(2*n-k,n-k-1)). - Vladimir Kruchinin, Oct 22 2016
D-finite with recurrence 2*(n+1)*a(n) -9*n*a(n-1) -3*n*a(n-2) +5*(5*n-16)*a(n-3) +6*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jul 22 2022
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EXAMPLE
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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).
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MAPLE
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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);
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MATHEMATICA
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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-François Alcover, Oct 22 2016, after Vladimir Kruchinin *)
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PROG
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(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
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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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