0,2

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

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.

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.

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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.

T. D. Noe, Table of n, a(n) for n=0..100

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]

(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 */

Cf. A000172, A002895, A000984.

Sequence in context: A020018 A124553 A020108 this_sequence A074539 A103210 A060066

Adjacent sequences: A002890 A002891 A002892 this_sequence A002894 A002895 A002896

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003