A106053 Next-to-central column of triangle in A059317.

0, 0, 1, 2, 8, 22, 72, 218, 691, 2158, 6833, 21612, 68726, 218892, 699197, 2237450, 7174018, 23038582, 74097134, 238625222, 769407486, 2483532218, 8024499657, 25951580444, 83999410292, 272098963300, 882045339733, 2861184745710, 9286923094550, 30161343633746

Offset: 0

Author

N. J. A. Sloane, May 28 2005

Keywords

• nonn

Comments

Number of h steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths hhh, hH, Hh, hUD, UhD and UDh we have altogether 8 h-steps. a(n) = Sum_{k=0..n-1} k*A132277(n-1,k). - Emeric Deutsch, Sep 03 2007

Number of paths in the right half-plane from (0,0) to (n-1,1) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because we have hhU, HU, hUh, Uhh, UH, DUU, UDU and UUD. Number of h-steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths from (0,0) to (3,0), namely, hhh, hH, Hh, hUD, UhD and UDh, we have altogether 8 h-steps. a(n) = Sum_{k=0..n-1} k*A132277(n-1,k). - 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, A128720, A132277.

Programs

Maple

g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2)): gser:=series(g,z=0,33); seq(coeff(gser,z,n),n=0..29); # Emeric Deutsch, Sep 03 2007
g:=((1-z-z^2)*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2))-1/2: gser:=series(g,z=0,33): 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 <= 2n := 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-2];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Aug 07 2018 *)

Formula

G.f.: (1 - z - z^2 - sqrt((1+z-z^2)*(1-3z-z^2)))/(2*sqrt((1+z-z^2)*(1-3z-z^2))). - Emeric Deutsch, Sep 03 2007

G.f.: (1-z-z^2)/(2*sqrt((1+z-z^2)*(1-3z-z^2))) - 1/2. - Emeric Deutsch, Sep 03 2007