OFFSET
0,3
COMMENTS
Number of paths in the right half-plane from (0,0) to (n,0) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(3)=9 because we have hhh, hH, Hh, hUD, hDU, UhD, DhU, UDh and DUh. The number of such paths restricted to the first quadrant is given in A128720. - Emeric Deutsch, Sep 03 2007
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), (2,2). - Joerg Arndt, Jun 30 2011
Other two columns of the triangle in A059317 are given in A106053 and A106050. - Emeric Deutsch, Sep 03 2007
REFERENCES
Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
LINKS
T. D. Noe, Table of n, a(n) for n=0..200
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.
FORMULA
G.f.: 1/sqrt((1+z-z^2)*(1-3*z-z^2)). - Emeric Deutsch, Sep 03 2007
D-finite with recurrence: (n+1)*a(n+1)=(2*n+1)*a(n)+5*n*a(n-1)-(2*n-1)*a(n-2)-(n-1)*a(n-3). - Emeric Deutsch, Sep 03 2007
a(n) = sum{k=0..floor(n/2), C(n-k,k)*A002426(n-2k)}. - Paul Barry, Nov 29 2008
G.f.: A(x) = Sum_{n>=0} (2*n)!/(n!)^2 * x^(2n)/(1-x-x^2)^(2n+1). - Paul D. Hanna, Oct 29 2010
a(n) ~ sqrt((3+11/sqrt(13))/8) * ((3+sqrt(13))/2)^n/sqrt(Pi*n). - Vaclav Kotesovec, Aug 11 2013
MAPLE
r:=proc(i, j) if i=0 then 0 elif i=1 and abs(j)>0 then 0 elif i=1 and j=0 then 1 elif i>=1 then r(i-1, j)+r(i-1, j-1)+r(i-1, j+1)+r(i-2, j) else 0 fi end: seq(r(i, 0), i=1..12); # very slow; Emeric Deutsch, Jun 06 2004
G:=1/sqrt((1+z-z^2)*(1-3*z-z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27); # Emeric Deutsch, Sep 03 2007
a[0]:=1: a[1]:=1: a[2]:=4: a[3]:=9: for n from 3 to 26 do a[n+1]:=((2*n+1)*a[n]+5*n*a[n-1]-(2*n-1)*a[n-2]-(n-1)*a[n-3])/(n+1) end do: seq(a[n], n=0..27); # Emeric Deutsch, Sep 03 2007
MATHEMATICA
CoefficientList[Series[1/Sqrt[(1+x-x^2)(1-3x-x^2)], {x, 0, 40}], x] (* Harvey P. Dale, Jun 04 2011 *)
a[n_] := Sum[Binomial[n-k, k]*Hypergeometric2F1[(2*k-n)/2, (2*k-n+1)/2, 1, 4], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 26 2015 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, x^(2*m)/(1-x-x^2+x*O(x^n))^(2*m+1)*(2*m)!/(m!)^2), n)} \\ Paul D. Hanna, Oct 29 2010
(PARI) /* same as in A092566 but use */
steps=[[1, 0], [1, 1], [1, 2], [2, 2]];
/* Joerg Arndt, Jun 30 2011 */
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jan 27 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001
STATUS
approved