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%I #13 Jul 01 2020 13:37:32
%S 2,15,69,252,804,2349,6455,16962,43086,106587,258153,614520,1441928,
%T 3342489,7667883,17432766,39321810,88080615,196083965,434110740,
%U 956301612,2097152325,4580180319,9965666682,21609054614,46707769779
%N Number of Dyck paths of semilength n+4, having exactly two long ascents (i.e., ascents of length at least two).
%C Also number of ordered trees with n+4 edges, having exactly two branch nodes (i.e., vertices of outdegree at least two).
%H Colin Barker, <a href="/A091135/b091135.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (9,-33,63,-66,36,-8).
%F a(n) = (n^2 + 9*n + 20)/2 + 2^(n+1)*(n^2 + 3*n - 4).
%F G.f.: (2 - 3*x)/((1 - 2*x)^3*(1 - x)^3).
%F 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
%e 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)).
%t LinearRecurrence[{9,-33,63,-66,36,-8},{2,15,69,252,804,2349},30] (* _Harvey P. Dale_, Jul 01 2020 *)
%o (PARI) Vec((2 - 3*x) / ((1 - x)^3*(1 - 2*x)^3) + O(x^40)) \\ _Colin Barker_, Apr 09 2019
%Y Cf. A000108.
%K nonn,easy
%O 0,1
%A _Emeric Deutsch_, Feb 22 2004
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