%I A059231
%S A059231 1,1,5,29,185,1257,8925,65445,491825,3768209,29324405,231153133,
%T A059231 1841801065,14810069497,120029657805,979470140661,8040831465825,
%U A059231 66361595715105,550284185213925,4582462506008253,38306388126997785
%N A059231 Number of different lattice paths running from (0,0) to (n,0) using steps 
               from S={(k,k) or (k,-k): k positive integer} that never go below 
               x-axis.
%C A059231 If y=x*A(x) then 4y^2-(1+3x)y+x=0 and x=y(1-4y)/(1-3y). - Michael Somos, 
               Sep 28 2003
%C A059231 a(n)=A059450(n,n). - Michael Somos Mar 06 2004
%C A059231 The Hankel transform of this sequence is 4^C(n+1,2) . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 29 2007
%D A059231 C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 
               13-28 (the sequence d_n).
%D A059231 C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
%D A059231 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the 
               Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 
               06.1.1.
%D A059231 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal 
               Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%F A059231 a(n)=sum(k=0, n, 4^k*N(n, k)) where N(n, k) = (1/n)*C(n, k)*C(n, k+1) 
               are the Narayana numbers (A001263) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 10 2003
%F A059231 a(n) = 3^n/2*LegendreP(n, -1, 5/3). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Sep 17 2003
%F A059231 G.f.: (1+3x-sqrt(1-10x+9x^2))/(8x)=2/(1+3x+sqrt(1-10x+9x^2)). (from Michael 
               Somos)
%F A059231 a(n) = Sum_{k=0..n} A088617(n, k)*4^k*(-3)^(n-k) . - DELEHAM Philippe 
               (kolotoko(AT)wanadoo.fr), Jan 21 2004
%F A059231 With offset 1 : a(1)=1, a(n)=-3*a(n-1)+4*sum(i=1, n-1, a(i)*a(n-i)) - 
               Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2004
%F A059231 a(n)=[5(2n-1)a(n-1)-9(n-2)a(n-2)]/(n+1) for n>=2; a(0)=a(1)=1. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Mar 20 2004
%F A059231 Moment representation: a(n)=(1/(8*pi))*Int(x^n*sqrt(-x^2+10x-9)/x,x,1,
               9)+(3/4)*0^n. [From Paul Barry (pbarry(AT)wit.ie), Sep 30 2009]
%p A059231 gf := (1+3*x-sqrt(9*x^2-10*x+1))/(8*x): s := series(gf, x, 100): for 
               i from 0 to 50 do printf(`%d,`,coeff(s, x, i)) od:
%o A059231 (PARI) a(n)=if(n<0,0,polcoeff((1+3*x-sqrt(1-10*x+9*x^2+x^2*O(x^n)))/(8*x),
               n)) (from Michael Somos)
%o A059231 (PARI) a(n)=if(n<0,0,n++; polcoeff(serreverse(x*(1-4*x)/(1-3*x)+x*O(x^ 
               n)),n)) (from Michael Somos)
%Y A059231 Cf. A001003, A007564.
%Y A059231 Row sums of A086873.
%Y A059231 Sequence in context: A153296 A153391 A081336 this_sequence A127846 A137573 
               A078945
%Y A059231 Adjacent sequences: A059228 A059229 A059230 this_sequence A059232 A059233 
               A059234
%K A059231 nonn,easy
%O A059231 0,3
%A A059231 Wen-jin Woan (wwoan(AT)fac.howard.edu), Jan 20 2001
%E A059231 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 22 2001