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

0,4

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

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.: (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

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 *)

Crossrefs

Cf. A059317, A128720, A132277.

Sequence in context: A089586 A339302 A045695 * A121135 A183410 A072929

Adjacent sequences: A106050 A106051 A106052 * A106054 A106055 A106056

Keyword

nonn

Author

N. J. A. Sloane, May 28 2005

Status

approved