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A082298
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G.f.: (1-3*x-sqrt(9*x^2-10*x+1))/(2*x).
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12
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1, 4, 20, 116, 740, 5028, 35700, 261780, 1967300, 15072836, 117297620, 924612532, 7367204260, 59240277988, 480118631220, 3917880562644, 32163325863300, 265446382860420, 2201136740855700, 18329850024033012, 153225552507991140
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OFFSET
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0,2
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COMMENTS
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More generally coefficients of (1-m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/(2*x) are given by a(0)=1 and, for n>0, a(n) = (1/n)*Sum_{k=0..n}(m+1)^k*binomial(n,k)*binomial(n,k-1).
a(n) = number of lattice paths from (0,0) to (n+1,n+1) that consist of steps (i,0) and (0,j) with i,j>=1 and that stay strictly below the diagonal line y=x except at the endpoints. (See Coker reference.) Equivalently, a(n) = number of marked Dyck (n+1)-paths where the vertices in the middle of each UU and each DD are available to be marked (or not): consider the original path as a Dyck path with a mark at each vertex where two horizontal (or two vertical) steps abut. If only the UU vertices are available for marking, then the counting sequence is the little Schroeder number A001003. - David Callan, Jun 07 2006
a(n) is the number of Schroder paths of semilength n in which the (2,0)-steps come in 3 colors. Example: a(2)=20 because, denoting U=(1,1), H=(2,0), D=(1,-1), we have 3^2=9 paths of shape HH, 3 paths of shape HUD, 3 paths of shape UDH, 3 paths of shape UHD, and 1 path of each of the shapes UDUD, UUDD. - Emeric Deutsch, May 02 2011
(1 + 4x + 20x^2 + 116x^3 + ...) = (1 + 5x + 29x^2 + 185x^3 + ...) * 1/(1 + x + 5x^2 + 29x^3 +185x^4 + ...); where A059231 = (1, 5, 29, 185, 1257, ...) - Gary W. Adamson, Nov 17 2011
The first differences between the row sums of the triangle A226392. - J. M. Bergot, Jun 21 2013
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LINKS
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FORMULA
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a(0)=1, n>0 a(n) = (1/n)*Sum_{k=0..n} 4^k*binomial(n, k)*binomial(n, k-1).
a(1)=1, a(n) = 3*a(n-1) + Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
a(n) = Sum_{k=0..n} 1/(n+1) Binomial(n+1,k)Binomial(2n-k,n-k)3^k. - David Callan, Jun 07 2006
G.f.: 1/(1-3x-x/(1-3x-x/(1-3x-x/(1-... (continued fraction);
a(n) = Sum_{k=0..n} binomial(n+k,2k)*3^(n-k)*A000108(k). (End)
D-finite with recurrence: (n+1)*a(n) = 5*(2n-1)*a(n-1)-9*(n-2)*a(n-2). - Paul Barry, Oct 22 2009
G.f.: 1/(1- 4x/(1-x/(1-4x/(1-x/(1-4x/(1-... (continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009
G.f.: (1-3*x-sqrt(9*x^2-10*x+1))/(2*x) = (1-G(0))/x; G(k) = 1+x*3-x*4/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012
a(n) = 4*hypergeom([1 - n, -n], [2], 4)) for n>0. - Peter Luschny, May 22 2017
G.f. A(x) satisfies: A(x) = (1 + x*A(x)^2) / (1 - 3*x). - Ilya Gutkovskiy, Jun 30 2020
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MAPLE
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A082298_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 4*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A082298_list(20); # Peter Luschny, May 19 2011
a := n -> `if`(n=0, 1, 4*hypergeom([1 - n, -n], [2], 4)):
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MATHEMATICA
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gf[x_] = (1 - 3*x - Sqrt[(9*x^2 - 10*x + 1)])/(2*x); CoefficientList[Series[gf[x], {x, 0, 20}], x] (* Jean-François Alcover, Jun 01 2011 *)
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PROG
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(PARI) a(n)=if(n<1, 1, sum(k=0, n, 4^k*binomial(n, k)*binomial(n, k-1))/n)
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-3*x-Sqrt(9*x^2-10*x+1))/(2*x))) // G. C. Greubel, Feb 10 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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