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A216318
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Number of peaks in all Dyck n-paths after changing each valley to a peak by the transform DU -> UD.
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1
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0, 1, 2, 8, 31, 119, 456, 1749, 6721, 25883, 99892, 386308, 1496782, 5809478, 22584160, 87922215, 342741285, 1337698515, 5226732060, 20442936360, 80031775890, 313585934610, 1229695855440, 4825705232010, 18950613058026, 74467158658974, 292797216620776, 1151895428382104
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: (16*x*(1+sqrt(1-4*x)+(5+3*sqrt(1-4*x)-2*x) * (-1+x)*x)) / ((1+sqrt(1-4*x))^5 * sqrt(1-4*x)).
a(n) = C(2*n-2,n-1)*(5*(n-1)^2+5*(n-1)+2)/(2*n*(n+1)), n>1, a(0)=0, a(1)=1. - Vladimir Kruchinin, Oct 30 2020
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EXAMPLE
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The 5 Dyck 3-paths after changing DU to UD become two copies of UUUDDD with one peak each and three copies of UUDUDD with two peaks each giving a(3)=8.
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MATHEMATICA
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CoefficientList[Series[(16*x*(1+Sqrt[1-4*x]+(5+3*Sqrt[1-4*x]-2*x)*(-1+x) x))/((1+Sqrt[1-4*x])^5*Sqrt[1-4*x]), {x, 0, 27}], x]
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PROG
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(PARI) x='x+O('x^50); concat([0], Vec((16*x*(1+sqrt(1-4*x)-(5+3*sqrt(1-4*x)-2*x)*(1-x)*x)) / ((1+sqrt(1-4*x))^5*sqrt(1-4*x)))) \\ G. C. Greubel, Apr 01 2017
(Maxima)
a(n):=if n<2 then n else binomial(2*n-2, n-1)*(5*(n-1)^2+5*(n-1)+2)/(2*n*(n+1)); /* Vladimir Kruchinin, Oct 30 2020 */
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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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