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%I #7 Jul 22 2022 08:34:37
%S 1,1,3,2,8,6,23,17,68,51,205,154,627,473,1937,1464,6032,4568,18900,
%T 14332,59519,45187,188211,143024,597241,454217,1900821,1446604,
%U 6065180,4618576,19396027
%N Number of symmetric paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0).
%C a(2n+1)=A059398(n); a(2n)=A059398(n-1)+A059398(n). The number of all paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) is A128720(n).
%F G.f.=2(1+z+z^2)/[1-3z^2-z^4+sqrt((1+z^2-z^4)(1-3z^2-z^4))].
%F D-finite with recurrence (n+2)*a(n) +n*a(n-1) +(-n-6)*a(n-2) -2*n*a(n-3) +7*(-n+2)*a(n-4) +5*(-n+4)*a(n-5) +3*(-n+6)*a(n-6) +2*(n-8)*a(n-7) +(3*n-26)*a(n-8) +(n-8)*a(n-9) +(n-10)*a(n-10)=0. - _R. J. Mathar_, Oct 08 2016
%e a(4)=8 because we have hhhh, hHh, HH, hUDh, UDUD, UhhD, UHD and UUDD.
%p G:=(2*(1+z+z^2))/(1-3*z^2-z^4+sqrt((1+z^2-z^4)*(1-3*z^2-z^4))): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..30);
%Y Cf. A128720, A059398.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Sep 05 2007
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