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A002894
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Binomial(2n,n)^2.
(Formerly M3664 N1490)
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34
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1, 4, 36, 400, 4900, 63504, 853776, 11778624, 165636900, 2363904400, 34134779536, 497634306624, 7312459672336, 108172480360000, 1609341595560000, 24061445010950400, 361297635242552100
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) is the number of monotonic paths (only moving N and E) in the lattice [0..2n] X [0..2n] that contain the points (0,0), (n,n) and (2n,2n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion of a special point on a curve described by Beauville.
Expansion of K(k)/(pi/2) in powers of m/16=(k/4)^2, where K(k) is complete elliptic integral of first kind evaluated at k. - Michael Somos, Mar 04 2003
Square lattice walks that start and end at origin after 2n steps. - Gareth McCaughan (gareth.mccaughan(AT)pobox.com) and Michael Somos Jun 12 2004
If A is a random matrix in USp(4) (4 X 4 X 4omplex matrices that are unitary and symplectic) then a(n)=E[(tr(A^k))^{2n}] for any k > 4. - Andrew V. Sutherland (drew(AT)math.mit.edu), Apr 01 2008
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591,828.
E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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.
Murray Elder, Cogrowth, http://carma.newcastle.edu.au/pdf/retreat2011/posters/MurrayElder-poster-A0.pdf.
Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
Eric M. Rains, High powers of random elements of compact Lie groups, Probability Theory and Related Fields 107 (1997), 219-241.
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
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. Bacher, Meander algebras
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 90
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.
L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| (n+1)^2 a_{n+1} = 16n^2 a_{n}. - Matthijs Coster, Apr 28, 2004
a(n) ~ pi^-1*n^-1*2^(4*n) - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
G.f.: F(1/2, 1/2;1;16x) = 1/AGM(1, (1-16x)^(1/2)) = K(4sqrt(x))/(pi/2), where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. - Michael Somos, Mar 04 2003
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2x)^2.
a(n) = A000984(n)^2 = ((2*n)!/(n!)^2)^2 = (((2*n)!)^2)/((n!)^4). a(n) = A000984(n)^2 = ((((2^n)*(2*n-1)!!)/(n!)))^2 = (((2^(2*n))*(2*n-1)!!)^2)/(n!)^2). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 17 2007
E.g.f.: (BesselI(0, 2x))^2=1+(2*x^2)/(U(0)-2*x^2); U(k)=(2*x^2)*(2*k+1)+(k+1)^3-(2*x^2)*(2*k+3)*((k+1)^3)/U(k+1)) ; (continued fraction). - Sergei N. Gladkovskii, Nov 23 2011
In generally, for (BesselI(b, 2x))^2=((x^(2*b))/(GAMMA(b+1))^2)*(1+(2*x^2)*(2*b+1)/(Q(0)-(2*x^2)*(2*b+1)); Q(k)=(2*x^2)*(2*k+2*b+1)+(k+1)*(k+b+1)*(k+2*b+1)-(2*x^2)*(k+1)*(k+b+1)*(k+2*b+1)*(2*k+2*b+3)/Q(k+1)) ; (continued fraction). - Sergei N. Gladkovskii, Nov 23 2011
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MAPLE
| A002894 := n-> binomial(2*n, n)^2.
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MATHEMATICA
| CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16x], {x, 0, 20}], x]
Table[Binomial[2n, n]^2, {n, 0, 20}] (* From Harvey P. Dale, July 06 2011 *)
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PROG
| (PARI) a(n)=binomial(2*n, n)^2
(PARI) a(n)=if(n<0, 0, polcoeff(polcoeff(polcoeff(1/(1-x*(y+z+1/y+1/z))+x*O(x^(2*n)), 2*n), 0), 0)) /* Michael Somos Jun 12 2004 */
(Other) sage: [binomial(2*n, n)**2 for n in xrange(0, 17)] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2009]
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CROSSREFS
| Cf. A000984, A000515, A010370, A054474, A060150.
Cf. A172390. [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 21 2010]
Sequence in context: A138736 A198638 A019999 * A202828 A131765 A132864
Adjacent sequences: A002891 A002892 A002893 * A002895 A002896 A002897
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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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