login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005573 Number of walks on cubic lattice (starting from origin and not going below xy plane).
(Formerly M3943)
12
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; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A026378, second binomial transform of A001700 . - Philippe Deléham, Jan 28 2007

The Hankel transform of [1,1,5,26,139,758,...] is [1,4,15,56,209,...](see A001353). - Philippe Deléham, Apr 13 2007

REFERENCES

E. Deutsch et al., Problem 10795, Amer. Math. Monthly, 108 (Dec. 2001), 980.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

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.

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. - Emeric Deutsch; corrected by Roland Bacher, 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, 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, Jul 16 2004

a(n) = Sum_{k, 0<=k<=n} A052179(n,k). - Philippe Deléham, Jan 28 2007

Conjecture: a(n)= 6^n * hypergeom([1/2,-n],[2], 2/3). - Benjamin Phillabaum, Feb 20 2011

From Paul Barry, 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). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Jul 02 2010

a(n) ~ 6^(n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 05 2012

G.f.: G(0)/(2*x) - 1/(2*x), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-2*x) - 2*x*(1-2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-2*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013

a(n) = 2^n*hypergeom([-n, 3/2], [2], -2). - Peter Luschny, Apr 26 2016

MATHEMATICA

CoefficientList[Series[(Sqrt[(1-2x)/(1-6x)]-1)/(2x), {x, 0, 20}], x] (* Harvey P. Dale, Jun 24 2011 *)

a[n_] := 6^n Hypergeometric2F1[1/2, -n, 2, 2/3]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 11 2017 *)

CROSSREFS

Sequence in context: A049607 A035029 A081569 * A081911 A081187 A182401

Adjacent sequences:  A005570 A005571 A005572 * A005574 A005575 A005576

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Henry Bottomley, Aug 23 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified April 28 00:46 EDT 2017. Contains 285556 sequences.