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A000888
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(2*n)!^2 / ((n+1)!*n!^3).
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13
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1, 2, 12, 100, 980, 10584, 121968, 1472328, 18404100, 236390440, 3103161776, 41469525552, 562496897872, 7726605740000, 107289439704000, 1503840313184400, 21252802073091300, 302539888334593800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) = number of walks of 2n unit steps North, East, South, or West, starting at the origin, bounded above by y=x, below by y=-x and terminating on the ray y=x>=0. Example: a(1) counts EN, EW; a(2) counts ESNN, ESNW, ENSN, ENSW, ENEN, ENEW, EENN, EENW, EEWN, EEWW, EWEN, EWEW. - David Callan, Oct 11 2005
Bijective proof: given such a NESW walk, construct a pair (P_1, P_2) of lattice paths of upsteps U=(1,1) and downsteps D=(1,-1) as follows. To get P_1, replace each E and S by U and each W and N by D. To get P_2, replace each N and E by U and each S and W by D. For example, EENSNW -> (UUDUDD, UUUDUD). This mapping is 1-to-1 and its range is the Cartesian product of the set of Dyck n-paths and the set of nonnegative paths of length 2n. The Dyck paths are counted by the Catalan number C_n (A000108) and the nonnegative paths are counted (see for example the Callan link) by the central binomial coefficient binom(2n,n) (A000984). So this is a bijection from these NESW walks to a set of size C_n*binom(2n,n)=a(n). - David Callan, Sep 18 2007
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n)=E[(tr(A^3))^{2n}]. - Andrew V. Sutherland (drew(AT)math.mit.edu), Apr 01 2008
a(n) is equal to the n-th moment of the following positive function defined on x in (0,16), in Maple notation: (EllipticK(sqrt(1-x/16)) - EllipticE(sqrt(1-x/16)))/(Pi^2*sqrt(x)). This is the solution of the Hausdorff moment problem and thus it is unique. - Karol A. Penson, Feb 11 2011.
a(n)= 2*A125558(n) (n>=1) [From Olivier Gerard, Feb 16 2011]
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REFERENCES
| E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 93.
Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
T. M. Macrobert, Functions of a Complex Variable, 4th ed., Macmillan & Co, London, 1958, p. 177
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..100
David Callan, Bijections for the identity 4^n = ...
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.
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FORMULA
| G.f.: 1/4*((16*x-1)*EllipticK(4*x^(1/2))+EllipticE(4*x^(1/2)))/x/Pi. - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 12 2003
Given G.f. A(x), y = x*A(x) satisfies y = y'' * (1 - 16*x) * x/4. - Michael Somos, Sep 11 2005
a(n) = Binomial(2*n,n)^2/(n+1) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 27 2006
G.f.: 2F1(1/2,1/2;2;16*x); [From Paul Barry, Sep 03 2008]
A002894(n)=(n+1)*a(n).
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EXAMPLE
| 1 + 2*x + 12*x^2 + 100*x^3 + 980*x^4 + 10584*x^5 + 121968*x^6 + ...
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MAPLE
| [seq(binomial(2*n, n)^2/(n+1), n=0..17)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 27 2006
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MATHEMATICA
| f[n_] := Binomial[2 n, n]^2/(n + 1); Array[f, 18, 0] (* RGWv *)
a[ n_] := SeriesCoefficient[ (1/8) (EllipticE[ 16 x] - (1 - 16 x) EllipticK[ 16 x]) / (Pi/2), {x, 0, n + 1}] (* Michael Somos, Jan 23 2012 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, (2*n)!^2 / n!^4 / (n+1)) /* Michael Somos, Sep 11 2005 */
(MAGMA) [(Factorial(2*n))^2/(Factorial(n))^4/(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 15 2011
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CROSSREFS
| Cf. A000108, A125558.
Sequence in context: A009816 A064370 A138421 * A151392 A079821 A124102
Adjacent sequences: A000885 A000886 A000887 * A000889 A000890 A000891
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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