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%I #14 Aug 07 2018 02:57:36
%S 0,0,0,1,3,13,42,146,476,1574,5122,16706,54256,176254,571954,1856245,
%T 6023681,19551939,63476314,206145075,669695819,2176401235,7075521724,
%U 23011145314,74864599954,243652588070,793264765396,2583532274289,8416929889967,27430452311513
%N Column two-from-center of triangle A059317.
%C Number of paths in the right-half-plane from (0,0) to (n-1,2) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=3 because we have hUU, UhU and UUh. - _Emeric Deutsch_, Sep 03 2007
%H W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="http://www.fq.math.ca/Scanned/35-4/klostermeyer.pdf">A Pascal rhombus</a>, Fibonacci Quarterly, 35 (1997), 318-328.
%H José L. Ramírez, <a href="http://arxiv.org/abs/1511.04577">The Pascal Rhombus and the Generalized Grand Motzkin Paths</a>, arXiv:1511.04577 [math.CO], 2015.
%F G.f.: z^3*g^2/sqrt((1+z-z^2)(1-3z-z^2)), where g=1+zg+z^2*g+z^2*g^2=[1-z-z^2-sqrt((1+z-z^2)(1-3z--z^2))]/(2z^2). - _Emeric Deutsch_, Sep 03 2007
%p g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: gser:=series(z^3*g^2/sqrt((1+z-z^2)*(1-3*z-z^2)),z=0,32): seq(coeff(gser,z,n),n=0..30); # _Emeric Deutsch_, Sep 03 2007
%t t[0, 0] = t[1, 0] = t[1, 1] = t[1, 2] = 1;
%t t[n_ /; n >= 0, k_ /; k >= 0] /; k <= 2 n := t[n, k] = t[n-1, k] + t[n-1, k-1] + t[n-1, k-2] + t[n-2, k-2];
%t t[n_, k_] /; n<0 || k<0 || k>2n = 0;
%t a[n_] := t[n-1, n-3];
%t Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Aug 07 2018 *)
%Y Cf. A059317, A059345, A106053.
%K nonn
%O 0,5
%A _N. J. A. Sloane_, May 28 2005
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