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A182542
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Second diagonal of triangle in A145879.
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2
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1, 8, 46, 232, 1093, 4944, 21778, 94184, 401930, 1698160, 7119516, 29666704, 123012781, 508019104, 2091005866, 8582372584, 35141476126, 143595498544, 585720020356, 2385430111024, 9701814930466, 39411044641888, 159926316674356, 648348726966672, 2626193752638388
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OFFSET
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3,2
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COMMENTS
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Sum of valley heights over all Dyck n-paths. - David Scambler, Oct 05 2012
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LINKS
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FORMULA
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G.f. appears to be (1-2*x-sqrt(1-4*x))^2/(4*x*(1-4*x)). - Mark van Hoeij, Apr 19 2013
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EXAMPLE
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Dyck 4-paths with nonzero valley heights are: UUUD(2)UDDD, UUUDD(1)UDD, UUD(1)UUDDD, UUD(1)UD(1)UDD, UUD(1)UDD(0)UD, and UD(0)UUD(1)UDD, with valley heights shown in parentheses, giving a(4) = 8. - David Scambler, Oct 05 2012
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MATHEMATICA
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a[n_] := 4^(n - 1) - n CatalanNumber[n];
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PROG
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(Maxima)
a(n):=2*sum((4^i*binomial(2*(n-i), n-i-2))/(n-i), i, 0, n-1); /* Vladimir Kruchinin, Mar 29 2019 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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