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 A091135 Number of Dyck paths of semilength n+4, having exactly two long ascents (i.e., ascents of length at least two). 1
 2, 15, 69, 252, 804, 2349, 6455, 16962, 43086, 106587, 258153, 614520, 1441928, 3342489, 7667883, 17432766, 39321810, 88080615, 196083965, 434110740, 956301612, 2097152325, 4580180319, 9965666682, 21609054614, 46707769779 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Also number of ordered trees with n+4 edges, having exactly two branch nodes (i.e., vertices of outdegree at least two). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8). FORMULA a(n) = (n^2 + 9*n + 20)/2 + 2^(n+1)*(n^2 + 3*n - 4). G.f.: (2 - 3*x)/((1 - 2*x)^3*(1 - x)^3). a(n) = 9*a(n-1) - 33*a(n-2) + 63*a(n-3) - 66*a(n-4) + 36*a(n-5) - 8*a(n-6) for n>5. - Colin Barker, Apr 09 2019 EXAMPLE a(0)=2 because the only Dyck paths of semilength 4 that have exactly two long ascents are UUDDUUDD and UUDUUDDD (here U=(1,1) and D=(1,-1)). MATHEMATICA LinearRecurrence[{9, -33, 63, -66, 36, -8}, {2, 15, 69, 252, 804, 2349}, 30] (* Harvey P. Dale, Jul 01 2020 *) PROG (PARI) Vec((2 - 3*x) / ((1 - x)^3*(1 - 2*x)^3) + O(x^40)) \\ Colin Barker, Apr 09 2019 CROSSREFS Cf. A000108. Sequence in context: A055206 A216614 A146757 * A056037 A125903 A268644 Adjacent sequences: A091132 A091133 A091134 * A091136 A091137 A091138 KEYWORD nonn,easy AUTHOR Emeric Deutsch, Feb 22 2004 STATUS approved

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Last modified June 1 03:55 EDT 2023. Contains 363068 sequences. (Running on oeis4.)