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A191321
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Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having only ascents of even length (an ascent is a maximal sequence of consecutive (1,1)-steps).
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1
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1, 1, 1, 1, 2, 3, 4, 5, 9, 14, 20, 27, 47, 74, 109, 153, 262, 415, 622, 894, 1516, 2410, 3653, 5335, 8988, 14323, 21883, 32330, 54213, 86543, 133004, 198229, 331233, 529462, 817432, 1226719, 2044151, 3270870, 5068346, 7648526, 12716872, 20365398, 31651555, 47984938, 79636493, 127621431
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OFFSET
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0,5
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LINKS
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FORMULA
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G.f.: g(z) = 1/(1-z-z^4*c^2), where c = 1+z^4*c^3 or, equivalently, c=(2*sqrt(3)/(3*z^2))*sin((1/3)*arcsin(3*z^2*sqrt(3)/2)) (see the explicit expression of g(z) in the Maple program).
Recurrence: 4*(n-1)*n*(n+1)*(99*n^3 - 1131*n^2 + 4170*n - 4928)*a(n) = 4*(n-1)*n*(99*n^4 - 1032*n^3 + 2913*n^2 - 152*n - 5348)*a(n-1) + 4*(n-1)*(99*n^5 - 1032*n^4 + 3543*n^3 - 4190*n^2 + 1600*n - 3360)*a(n-2) - 24*(11*n^2 - 67*n + 96)*(21*n^2 - 101*n + 70)*a(n-3) + 3*(891*n^6 - 15525*n^5 + 106425*n^4 - 359079*n^3 + 600268*n^2 - 416180*n + 44800)*a(n-4) - 3*(891*n^6 - 15525*n^5 + 103905*n^4 - 329655*n^3 + 480420*n^2 - 222756*n - 49280)*a(n-5) - 3*n*(3*n - 14)*(3*n - 13)*(99*n^3 - 834*n^2 + 2205*n - 1790)*a(n-6). - Vaclav Kotesovec, Mar 17 2014
a(n) ~ (13+m) * 3^(3*n/4+m/4+1/2)/ (sqrt(Pi) * 2^(n/2-3+m/2) * n^(3/2)), where m = mod(n,4). - Vaclav Kotesovec, Mar 17 2014
a(n) = Sum_{m=0..n} ((m+1)*Sum_{i=0..floor((n+1)/4)-m-1)} ((binomial(2*m+3*i+1,i)*( binomial(n-3*(m+1)-4*i,m+1)))/(m+i+1)))+1. - Vladimir Kruchinin, Mar 12 2016
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EXAMPLE
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a(5)=3 because we have HHHHH, HUUDD, and UUDDH, where U=(1,1), H=(1,0), and D=(1,-1).
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MAPLE
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g := 3/(4*cos((1/3)*arcsin(3*z^2*sqrt(3)*1/2))^2-1-3*z): gser := series(g, z = 0, 48): seq(coeff(gser, z, n), n = 0 .. 45);
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MATHEMATICA
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CoefficientList[Series[3/(4*Cos[1/3*ArcSin[3*x^2*Sqrt[3]*1/2]]^2-1-3*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 17 2014 *)
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PROG
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(Maxima)
a(n):=sum((m+1)*sum((binomial(2*m+3*i+1, i)*( binomial(n-3*(m+1)-4*i, m+1)))/(m+i+1), i, 0, floor((n+1)/4)-(m+1)), m, 0, n)+1; /* Vladimir Kruchinin, Mar 11 2016 */
(PARI) a(n) = 1+ sum(m=0, n, ((m+1)*sum(i=0, floor((n+1)/4)-m-1, ((binomial(2*m+3*i+1, i)*( binomial(n-3*(m+1)-4*i, m+1)))/(m+i+1))))); \\ Michel Marcus, Mar 12 2016
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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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