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

0,5

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

Table of n, a(n) for n=0..29.

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.

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

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 *)

Crossrefs

Cf. A059317, A059345, A106053.

Sequence in context: A146018 A145946 A109224 * A074425 A041499 A093923

Adjacent sequences: A106047 A106048 A106049 * A106051 A106052 A106053

Keyword

nonn

Author

N. J. A. Sloane, May 28 2005

Status

approved