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A167638
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Number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having no peaks at even level.
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2
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1, 0, 0, 1, 0, 2, 1, 5, 5, 15, 21, 51, 85, 188, 344, 730, 1407, 2935, 5831, 12094, 24480, 50754, 103995, 216043, 446447, 930206, 1934328, 4043275, 8448882, 17716170, 37166403, 78163336, 164520540, 346935912, 732317063, 1548096255, 3275859473
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OFFSET
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0,6
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1 + 2z - z^3 - sqrt(1 - 4z^2 - 2z^3 + z^6))/(2z(1 + z )).
a(n) = Sum_{i=0..n} Sum_{k=1..n-i} Sum_{j=0..i-k+1} (-1)^(n-j-1)*C(j,-k-j+i+1)*C(k+j-1,k-1)*C(2*k+j-2,k+j-1)*C(n-i-1,n-k-i))/k. - Vladimir Kruchinin, May 06 2018
D-finite with recurrence (n+1)*a(n) +a(n-1) +(-4*n+7)*a(n-2) +2*(-n+2)*a(n-3) +a(n-4) -a(n-5) +(n-7)*a(n-6)=0. - R. J. Mathar, Jul 24 2022
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EXAMPLE
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a(5)=2 because we have UUUDDUUDDD and UUUUUDDDDD.
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MAPLE
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G := ((1+2*z-z^3-sqrt(1-4*z^2-2*z^3+z^6))*1/2)/(z*(1+z)): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 38);
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PROG
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(Maxima)
a(n):=sum(sum(((sum((-1)^(n-j-1)*binomial(j, -k-j+i+1)*binomial(k+j-1, k-1)*binomial(2*k+j-2, k+j-1), j, 0, -k+i+1))*binomial(n-i-1, n-k-i))/k, k, 1, n-i), i, 0, n);
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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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