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A002895
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Number of 2n-step polygons on diamond lattice.
(Formerly M3626 N1473)
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4
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1, 4, 28, 256, 2716, 31504, 387136, 4951552, 65218204, 878536624, 12046924528, 167595457792, 2359613230144, 33557651538688, 481365424895488, 6956365106016256, 101181938814289564, 1480129751586116848
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) is the (2n)th moment of the distance from the origin of a 4-step random walk in the plane - Peter M.W. Gill (peter.gill(AT)nott.ac.uk), Mar 03 2004
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REFERENCES
| David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, http://carma.newcastle.edu.au/~jb616/wmi-paper.pdf.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
J. A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
L. B. Richmond, J. Shallit, Counting Abelian Squares, arXiv:0807.5028 [Math.CO]. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 30 2008]
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FORMULA
| Sum_{k=0..n} binomial(n, k)^2 binomial(2n-2k, n-k) binomial(2k, k).
n^3*a(n) = 2*(2*n-1)*(5*n^2-5*n+2)*a(n-1)-64*(n-1)^3*a(n-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 16 2004
Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0, 2*sqrt(x))^4. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 01 2006
G.f.: hypergeom([1/6, 1/3],[1],108*x^2/(1-4*x)^3)^2/(1-4*x) - Mark van Hoeij, Oct 29 2011.
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MATHEMATICA
| Table[Sum[Binomial[n, k]^2 Binomial[2n-2k, n-k]Binomial[2k, k], {k, 0, n}], {n, 0, 30}] (* From Harvey P. Dale, Aug 15 2011 *)
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CROSSREFS
| Cf. A002893.
Sequence in context: A103211 A191801 A064340 * A141004 A152410 A177403
Adjacent sequences: A002892 A002893 A002894 * A002896 A002897 A002898
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KEYWORD
| nonn,easy,nice,walk
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 11 2003
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