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A098430
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4^n*(2*n)!/(n!)^2.
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3
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1, 8, 96, 1280, 17920, 258048, 3784704, 56229888, 843448320, 12745441280, 193730707456, 2958796259328, 45368209309696, 697972450918400, 10768717814169600, 166556168859156480, 2581620617316925440
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 4^n binom[2n,n] counts walks of 2n steps North, East, South or West that start at the origin and end on the line y=x. For example, a(1)=8 counts EW, EN, NE, NS, WE, WS, SN, SW. If the walk has i East and j North steps, then it must have n-j West and n-i South steps. There are Multinomial[i,j,n-j,n-i] ways to arrange these steps and summing over i and j gives the result. - David Callan (callan(AT)stat.wisc.edu), Oct 11 2005
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), both of two kinds. [Joerg Arndt, Jul 01 2011]
Hankel transform is A121913. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 01 2009]
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REFERENCES
| Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
| a(n) = 4^n*Binomial(2*n, n).
E.g.f.: exp(8*x)*BesselI(0, 8*x).
a(n)=4^n*A000984(n).
G.f.:1/sqrt(1-16*x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 20 2008, corrected R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 18 2009]
a(n) = (1/Pi)*integral(x=-2..2, (2*x)^(2*n)/(sqrt((2-x)*(2+x))). [Peter Luschny, Sep 12 2011]
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PROG
| (PARI) /* as lattice paths: same as in A092566 but use */
steps=[[1, 0], [1, 0], [0, 1], [0, 1]]; /* note the double [1, 0] and [0, 1] */
/* Joerg Arndt, Jul 01 2011 */
(MAGMA) [4^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011
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CROSSREFS
| Sequence in context: A099675 A060458 A173834 * A034177 A052570 A002168
Adjacent sequences: A098427 A098428 A098429 * A098431 A098432 A098433
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 07 2004
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