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A002893
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Sum_{k=0..n} binomial(n,k)^2 * binomial(2k,k).
(Formerly M2998 N1214)
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13
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1, 3, 15, 93, 639, 4653, 35169, 272835, 2157759, 17319837, 140668065, 1153462995, 9533639025, 79326566595, 663835030335, 5582724468093, 47152425626559, 399769750195965, 3400775573443089, 29016970072920387, 248256043372999089
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion of a special point on a curve described by Beauville.
a(n) is the (2n)th moment of the distance from the origin of a 3-step random walk in the plane - Peter M. W. Gill (peter.gill(AT)nott.ac.uk), Feb 27 2004
a(n) is the number of Abelian squares of length 2n over a 3-letter alphabet. [From Jeffrey Shallit (shallit(AT)cs.uwaterloo.ca), Aug 17 2010]
Consider 2D simple random walk on honeycomb lattice. a(n) gives number of paths of length 2n ending at origin - Sergey Perepechko (persn(AT)aport.ru) Feb 16 2011
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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.
P. Barrucand, A combinatorial identity, Problem 75-4, SIAM Rev., 17 (1975), 168.
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, http://carma.newcastle.edu.au/~jb616/wmi-paper.pdf.
Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
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.
M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 148-149.
L. B. Richmond and J. Shallit, Counting abelian squares, Electronic J. Combinatorics 16 (1), #R72, June 2009. [From Jeffrey Shallit (shallit(AT)cs.uwaterloo.ca), Aug 17 2010]
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
| T. D. Noe, Table of n, a(n) for n=0..100
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
L. B. Richmond and J. Shallit, Counting abelian squares, Electronic J. Combinatorics 16 (1), #R72, June 2009. [From Jeffrey Shallit (shallit(AT)cs.uwaterloo.ca), Aug 17 2010]
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FORMULA
| a(n) = Sum_{m=0..n} binomial(n, m) A000172(m) [Barrucand]
(n+1)^2 a_{n+1} = (10n^2+10n+3) a_{n} - 9n^2 a_{n-1}. - Matthijs Coster, Apr 28, 2004
Sum_{n>=0} a(n)x^n/n!^2 = BesselI(0, 2*sqrt(x))^3. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 11 2003
a(n) = Sum_{p+q+r=n} (n!/(p!q!r!))^2 with p,q,r >=0. - Michael Somos Jul 25 2007
a(n)=3*A087457(n)for n>0 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2008]
a(n) = hypergeom([1/2, -n, -n], [1, 1], 4) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Jun 02 2010]
Ordinary generating function - Sergey Perepechko (persn(AT)aport.ru) Feb 16 2011: Gf(z)=2*sqrt(2)/Pi/sqrt(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z))) * EllipticK(8*z^(3/2)/(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z))))
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MATHEMATICA
| Table[Sum[Binomial[n, k]^2 Binomial[2k, k], {k, 0, n}], {n, 0, 20}] (* From Harvey P. Dale, Aug 19 2011 *)
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PROG
| (PARI) a(n)=if(n<0, 0, n!^2*polcoeff(besseli(0, 2*x+O(x^(2*n+1)))^3, 2*n))
(PARI) {a(n)= sum(k=0, n, binomial(n, k)^2 * binomial(2*k, k))} /* Michael Somos Jul 25 2007 */
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CROSSREFS
| Cf. A000172, A002895, A000984.
Sequence in context: A020108 A193661 A192296 * A074539 A103210 A203014
Adjacent sequences: A002890 A002891 A002892 * A002894 A002895 A002896
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
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