A106050 Column two-from-center of triangle A059317.

0, 0, 0, 1, 3, 13, 42, 146, 476, 1574, 5122, 16706, 54256, 176254, 571954, 1856245, 6023681, 19551939, 63476314, 206145075, 669695819, 2176401235, 7075521724, 23011145314, 74864599954, 243652588070, 793264765396, 2583532274289, 8416929889967, 27430452311513

Offset: 0

Author

N. J. A. Sloane, May 28 2005

Keywords

• nonn

Comments

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

Links

• W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
• José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015.

Crossrefs

Cf. A059317, A059345, A106053.

Programs

Maple

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

Mathematica

t[0, 0] = t[1, 0] = t[1, 1] = t[1, 2] = 1;
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[n_, k_] /; n<0 || k<0 || k>2n = 0;
a[n_] := t[n-1, n-3];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Aug 07 2018 *)

Formula

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