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Search: id:A059231
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| A059231 |
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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. |
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17
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| 1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405, 231153133, 1841801065, 14810069497, 120029657805, 979470140661, 8040831465825, 66361595715105, 550284185213925, 4582462506008253, 38306388126997785
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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
a(n)=A059450(n,n). - Michael Somos Mar 06 2004
The Hankel transform of this sequence is 4^C(n+1,2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2007
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REFERENCES
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C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28 (the sequence d_n).
C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
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.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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FORMULA
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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
a(n) = 3^n/2*LegendreP(n, -1, 5/3). - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 17 2003
G.f.: (1+3x-sqrt(1-10x+9x^2))/(8x)=2/(1+3x+sqrt(1-10x+9x^2)). (from Michael Somos)
a(n) = Sum_{k=0..n} A088617(n, k)*4^k*(-3)^(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 21 2004
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
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
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]
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MAPLE
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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:
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PROGRAM
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(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)
(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)
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CROSSREFS
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Cf. A001003, A007564.
Row sums of A086873.
Sequence in context: A153296 A153391 A081336 this_sequence A127846 A137573 A078945
Adjacent sequences: A059228 A059229 A059230 this_sequence A059232 A059233 A059234
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KEYWORD
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nonn,easy
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AUTHOR
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Wen-jin Woan (wwoan(AT)fac.howard.edu), Jan 20 2001
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 22 2001
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