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A005573
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Number of walks on cubic lattice (starting from origin and not going below xy plane).
(Formerly M3943)
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11
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1, 5, 26, 139, 758, 4194, 23460, 132339, 751526, 4290838, 24607628, 141648830, 817952188, 4736107172, 27487711752, 159864676803, 931448227590, 5435879858958, 31769632683132, 185918669183370, 1089302293140564
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Binomial transform of A026378, second binomial transform of A001700 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 28 2007
The Hankel transform of [1,1,5,26,139,758,...] is [1,4,15,56,209,...](see A001353). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 13 2007
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REFERENCES
| E. Deutsch et al., Problem 10795, Amer. Math. Monthly, 108 (Dec. 2001), 980.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeT
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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FORMULA
| sum((-1)^i*6^(n-i)*binomial(n, i)*binomial(2*i, i)/(i+1), i=0..n); g.f. A(x) satisfies x(1-6x)A^2+(1-6x)A-1=0 - from Emeric Deutsch (deutsch(AT)duke.poly.edu); corrected by Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Jan 09 2003
a(n) = 6a(n-1)-A005572(n-1) = sum{j = 0, ..., n}[4^(n-j)*C(n, [n/2])*C(n, j)] - Henry Bottomley (se16(AT)btinternet.com), Aug 23 2001
a(n) = sum_{k=0..n} binomial(n, k)*binomial(2*k+1, k)*2^(n-k).
a(n) = sum_{k=0..n} (-1)^k*binomial(n, k)*Catalan(k)*6^(n-k).
(n+1)*a(n) = (8*n+2)*a(n-1)-(12*n-12)*a(n-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 16 2004
a(n) = Sum_{k, 0<=k<=n} A052179(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 28 2007
Conjecture: a(n)= (6)^n GaussHypergeometric(1/2,-n;2;2/3) - Benjamin Phillabaum
(bphillab (AT) gmail.com), Feb 20 2011
Contribution from Paul Barry (pbarry(AT)wit.ie), Apr 21 2009: (Start)
G.f.: (sqrt((1-2x)/(1-6x))-1)/(2x);
G.f.: 1/(1-5x-x^2/(1-4x-x^2/(1-4x-x^2/(1-4x-x^2/(1-... (continued fraction). (End)
G.f.: 1/(1 - 4x - x(1 - 2x)/(1 - 2x - x(1 - 2x)/(1 - 2x - x(1 - 2x)/(1 - 2x - x(1 - 2x)/(1...(continued fraction). [From Aoife Hennessy (aoife.hennessy(AT)gmail.com), Jul 02 2010]
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MATHEMATICA
| CoefficientList[Series[(Sqrt[(1-2x)/(1-6x)]-1)/(2x), {x, 0, 20}], x] (* From Harvey P. Dale, June 24 2011 *)
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CROSSREFS
| Sequence in context: A049607 A035029 A081569 * A081911 A081187 A104498
Adjacent sequences: A005570 A005571 A005572 * A005574 A005575 A005576
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Henry Bottomley (se16(AT)btinternet.com), Aug 23 2001
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