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A212694 Number of 2-colored Dyck n-paths with up steps (U, u), down steps (D, d), and avoiding UU and DD.

%I A212694 #25 Jun 02 2018 10:35:37
%S A212694 1,4,25,197,1745,16580,165115,1700809,17971466,193710087,2121585340,
%T A212694 23543198588,264138223362,2991130956918,34143543312267,
%U A212694 392458689992396,4538574332686469,52768896995910303,616471818881678085,7232838546289017796,85188401983572928395
%N A212694 Number of 2-colored Dyck n-paths with up steps (U, u), down steps (D, d), and avoiding UU and DD.
%C A212694 Upper case letters denote one color and lower case letters the other.
%H A212694 Alois P. Heinz, <a href="/A212694/b212694.txt">Table of n, a(n) for n = 0..910</a>
%H A212694 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A212694 Recurrence: 5*n*(n+1)*(n+2)*(11856*n^6 - 462048*n^5 + 6376819*n^4 - 42412433*n^3 + 147659510*n^2 - 260432089*n + 184927095)*a(n) = n*(n+1)*(1683552*n^7 - 66428880*n^6 + 937628962*n^5 - 6466755025*n^4 + 23922419618*n^3 - 47274340850*n^2 + 44490285903*n - 13108829940)*a(n-1) - n*(14310192*n^8 - 585624672*n^7 + 8802419365*n^6 - 66744441981*n^5 + 284660409448*n^4 - 703230360993*n^3 + 976943147665*n^2 - 682900257024*n + 175305383460)*a(n-2) + 2*(17238624*n^9 - 746332752*n^8 + 12295273466*n^7 - 105941663163*n^6 + 537013083761*n^5 - 1677467513328*n^4 + 3243096592679*n^3 - 3743377415442*n^2 + 2332897216785*n - 592736810400)*a(n-3) - (37974768*n^9 - 1728832752*n^8 + 30714335273*n^7 - 291055104422*n^6 + 1652510618897*n^5 - 5890634883203*n^4 + 13267453974662*n^3 - 18291224811263*n^2 + 14073816074160*n - 4638445144200)*a(n-4) + (23308896*n^9 - 1108835760*n^8 + 20950004098*n^7 - 213362099351*n^6 + 1311302140489*n^5 - 5085585201440*n^4 + 12508042515937*n^3 - 18889708965719*n^2 + 15984167161110*n - 5834960787600)*a(n-5) - (8927568*n^9 - 442319616*n^8 + 8817227331*n^7 - 95321285525*n^6 + 623567997102*n^5 - 2575734547859*n^4 + 6742249614729*n^3 - 10822975984520*n^2 + 9730758446550*n - 3782772932400)*a(n-6) + 4*(2*n - 13)*(248976*n^8 - 11244288*n^7 + 197289135*n^6 - 1812121625*n^5 + 9661474889*n^4 - 30841990357*n^3 + 57959845045*n^2 - 59304520365*n + 25796647800)*a(n-7) - 4*(n-7)*(2*n - 15)*(2*n - 13)*(11856*n^6 - 390912*n^5 + 4244419*n^4 - 21288517*n^3 + 54240485*n^2 - 69082196*n + 35668710)*a(n-8). - _Vaclav Kotesovec_, Jul 16 2014
%F A212694 Limit n->infinity a(n)^(1/n) = (13+3*sqrt(17))/2 = 12.68465843842649... . - _Vaclav Kotesovec_, Jul 16 2014
%e A212694 a(1) = 4: ud, Ud, uD, UD.
%e A212694 a(2) = 25: uudd, Uudd, uUdd, uuDd, UuDd, uUDd, udud, Udud, uDud, UDud, udUd, UdUd, uDUd, UDUd, uudD, UudD, uUdD, uduD, UduD, uDuD, UDuD, udUD, UdUD, uDUD, UDUD.
%p A212694 b:= proc(x, y, t) option remember; `if`(x=0, 1,
%p A212694       `if`(y<1  , 0, b(x-1, y-1, 0)+`if`(t=1, 0, b(x-1, y-1, 1)))+
%p A212694       `if`(x<y+2, 0, b(x-1, y+1, 0)+`if`(t=2, 0, b(x-1, y+1, 2))))
%p A212694     end:
%p A212694 a:= n-> b(2*n, 0$2):
%p A212694 seq(a(n), n=0..30);
%t A212694 b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1, 0] + If[t == 1, 0, b[x - 1, y - 1, 1]]] + If[x < y + 2, 0, b[x - 1, y + 1, 0] + If[t == 2, 0, b[x - 1, y + 1, 2]]]];
%t A212694 a[n_] := b[2n, 0, 0];
%t A212694 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A212694 Cf. A000108, A007863, A151403.
%K A212694 nonn
%O A212694 0,2
%A A212694 _Alois P. Heinz_, May 23 2012