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A059345 Central column of Pascal's "rhombus" (actually a triangle) A059317. 6
1, 1, 4, 9, 29, 82, 255, 773, 2410, 7499, 23575, 74298, 235325, 747407, 2381126, 7603433, 24332595, 78013192, 250540055, 805803691, 2595158718, 8368026845, 27012184877, 87283372610, 282294378071, 913775677281, 2960160734818 (list; graph; refs; listen; history; text; internal format)
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

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

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

Rec. rel.: (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-Fran├žois 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

Cf. A128720, A106050, A106053.

Cf. A181545. - Paul D. Hanna, Oct 29 2010

Sequence in context: A069563 A276984 A210969 * A127768 A231255 A241393

Adjacent sequences:  A059342 A059343 A059344 * A059346 A059347 A059348

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

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Last modified December 16 06:10 EST 2018. Contains 318158 sequences. (Running on oeis4.)