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A106050 Column two-from-center of triangle A059317. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 9 15:04 EST 2019. Contains 329877 sequences. (Running on oeis4.)