|
%I #42 Jun 07 2023 09:40:45
%S 1,1,4,9,29,82,255,773,2410,7499,23575,74298,235325,747407,2381126,
%T 7603433,24332595,78013192,250540055,805803691,2595158718,8368026845,
%U 27012184877,87283372610,282294378071,913775677281,2960160734818
%N Central column of Pascal's "rhombus" (actually a triangle) A059317.
%C 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
%C 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
%C Other two columns of the triangle in A059317 are given in A106053 and A106050. - _Emeric Deutsch_, Sep 03 2007
%D Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
%H T. D. Noe, <a href="/A059345/b059345.txt">Table of n, a(n) for n=0..200</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%H J. Goldwasser et al., <a href="http://dx.doi.org/10.1016/S0012-365X(98)00373-2">The density of ones in Pascal's rhombus</a>, Discrete Math., 204 (1999), 231-236.
%H Paul K. Stockmeyer, <a href="http://arxiv.org/abs/1504.04404">The Pascal Rhombus and the Stealth Configuration</a>, arXiv:1504.04404 [math.CO], 2015.
%F G.f.: 1/sqrt((1+z-z^2)*(1-3*z-z^2)). - _Emeric Deutsch_, Sep 03 2007
%F 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
%F a(n) = sum{k=0..floor(n/2), C(n-k,k)*A002426(n-2k)}. - _Paul Barry_, Nov 29 2008
%F 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
%F a(n) ~ sqrt((3+11/sqrt(13))/8) * ((3+sqrt(13))/2)^n/sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 11 2013
%p 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
%p 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
%p 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
%t CoefficientList[Series[1/Sqrt[(1+x-x^2)(1-3x-x^2)],{x,0,40}],x] (* _Harvey P. Dale_, Jun 04 2011 *)
%t 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 *)
%o (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
%o (PARI) /* same as in A092566 but use */
%o steps=[[1,0], [1,1], [1,2], [2,2]];
%o /* _Joerg Arndt_, Jun 30 2011 */
%Y Cf. A128720, A106050, A106053.
%Y Cf. A181545. - _Paul D. Hanna_, Oct 29 2010
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, Jan 27 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001
|
