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A106053 Next-to-central column of triangle in A059317. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A137103 A089586 A045695 * A121135 A183410 A072929

Adjacent sequences:  A106050 A106051 A106052 * A106054 A106055 A106056

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, May 28 2005

STATUS

approved

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Last modified December 6 19:31 EST 2019. Contains 329809 sequences. (Running on oeis4.)