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Search: id:A002895
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| A002895 |
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Number of 2n-step polygons on diamond lattice.
(Formerly M3626 N1473)
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+0
3
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| 1, 4, 28, 256, 2716, 31504, 387136, 4951552, 65218204, 878536624, 12046924528, 167595457792, 2359613230144, 33557651538688, 481365424895488, 6956365106016256, 101181938814289564, 1480129751586116848
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.
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.
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LINKS
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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
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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
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CROSSREFS
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Cf. A002893.
Sequence in context: A112113 A103211 A064340 this_sequence A141004 A152410 A138272
Adjacent sequences: A002892 A002893 A002894 this_sequence A002896 A002897 A002898
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KEYWORD
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nonn,easy,nice,walk
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 11 2003
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