%I A002893 M2998 N1214
%S A002893 1,3,15,93,639,4653,35169,272835,2157759,17319837,140668065,1153462995,
%T A002893 9533639025,79326566595,663835030335,5582724468093,47152425626559,
%U A002893 399769750195965,3400775573443089,29016970072920387,248256043372999089
%N A002893 Sum_{k=0..n} binomial(n,k)^2 * binomial(2k,k).
%C A002893 Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion 
               of a special point on a curve described by Beauville.
%C A002893 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
%D A002893 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002893 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002893 David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, 
               Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.
%D A002893 P. Barrucand, A combinatorial identity, Problem 75-4, SIAM Rev., 17 (1975), 
               168.
%D A002893 Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre 
               fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, 
               May 24 1982.
%D A002893 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], 
               Master's Thesis (unpublished), Aug 26 1983.
%D A002893 C. Domb, On the theory of cooperative phenomena in crystals, Advances 
               in Phys., 9 (1960), 149-361.
%D A002893 J. A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, 
               Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
%D A002893 M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM 
               Review, SIAM, 1990; see pp. 148-149.
%H A002893 T. D. Noe, <a href="b002893.txt">Table of n, a(n) for n=0..100</a>
%F A002893 a(n) = Sum_{m=0..n} binomial(n, m) A000172(m) [Barrucand]
%F A002893 (n+1)^2 a_{n+1} = (10n^2+10n+3) a_{n} - 9n^2 a_{n-1}. - Matthijs Coster, 
               Apr 28, 2004
%F A002893 Sum_{n>=0} a(n)x^n/n!^2 = BesselI(0, 2*sqrt(x))^3. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Mar 11 2003
%F A002893 a(n) = Sum_{p+q+r=n} (n!/(p!q!r!))^2 with p,q,r >=0. - Michael Somos 
               Jul 25 2007
%F A002893 a(n)=3*A087457(n)for n>0 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 14 2008]
%o A002893 (PARI) a(n)=if(n<0,0,n!^2*polcoeff(besseli(0,2*x+O(x^(2*n+1)))^3,2*n))
%o A002893 (PARI) {a(n)= sum(k=0, n, binomial(n, k)^2 * binomial(2*k, k))} /* Michael 
               Somos Jul 25 2007 */
%Y A002893 Cf. A000172, A002895, A000984.
%Y A002893 Sequence in context: A020018 A124553 A020108 this_sequence A074539 A103210 
               A060066
%Y A002893 Adjacent sequences: A002890 A002891 A002892 this_sequence A002894 A002895 
               A002896
%K A002893 nonn,easy,nice
%O A002893 0,2
%A A002893 N. J. A. Sloane (njas(AT)research.att.com).
%E A002893 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 
               29 2003