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A166302
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Sum of pyramid weights of all Dyck paths of semilength n that have no ascents and no descents of length 1.
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2
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0, 0, 2, 3, 8, 19, 44, 106, 257, 628, 1549, 3844, 9588, 24020, 60391, 152298, 385085, 975904, 2478129, 6303861, 16060946, 40977605, 104682165, 267730426, 685451776, 1756593392, 4505537267, 11565724164, 29711413595, 76379060176, 196473781247
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OFFSET
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0,3
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COMMENTS
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A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} k*A166301(n,k).
G.f.: z*(2-z)*[1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2))]/[2*(1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))].
a(n) ~ (3+sqrt(5))^(n+1/2) / (5^(1/4) * sqrt(Pi*n) * 2^(n+3/2)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence 2*(n-1)*(2494*n-8185)*a(n) +23*(-770*n^2+3867*n-3959)*a(n-1) +(13226*n^2-83741*n+101091)*a(n-2) +(-7734*n^2+51213*n-51521)*a(n-3) +(17710*n^2-114385*n+144471)*a(n-4) +(-13226*n^2+104701*n-162397)*a(n-5) +(2746*n-6149)*(n-7)*a(n-6)=0. - R. J. Mathar, Jul 24 2022
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EXAMPLE
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a(5)=19 because the pyramid weights of (UUDD)(UUUDDD), (UUUDDD)(UUDD), U(UUDD)(UUDD)D, and (UUUUUDDDDD) are 5, 5, 4, and 5, respectively (the maximal pyramids are shown between parentheses).
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MAPLE
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G := (1/2)*z*(2-z)*(1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/((1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
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MATHEMATICA
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CoefficientList[Series[1/2*x*(2-x)*(1+x-x^2-Sqrt[(1+x+x^2)*(1-3*x+x^2)]) /((1-x)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
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PROG
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(PARI) z='z+O('z^50); concat([0, 0], Vec(z*(2-z)*(1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/(2*(1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))))) \\ G. C. Greubel, Mar 22 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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