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A001246
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Squares of Catalan numbers.
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21
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1, 1, 4, 25, 196, 1764, 17424, 184041, 2044900, 23639044, 282105616, 3455793796, 43268992144, 551900410000, 7152629313600, 93990019574025, 1250164827828900, 16807771574144100, 228138727737690000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also multi-component meanders.
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REFERENCES
| O. Guibert, Stack words, ..., Discr. Math., 210 (2000), 71-85.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics.
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FORMULA
| G.f.: -1/(4*x)+1/2*(16*x-1)/x * EllipticK(4*x^(1/2))/Pi + 1/x*EllipticE(4*x^(1/2))/Pi. - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 12 2003
G.f.: 3F2( (1, 1/2, 1/2); (2, 2); 16x) = (-1 + 2F1( (-1/2, -1/2); (1); 16x))/(4*x) [From Olivier Gerard (olivier.gerard(AT)gmail.com), Feb 16 2011]
E.g.f.: hypergeom([1/2], [2, 2], 4*x^2) = 2*BesselI(0, 2*x)^2-BesselI(0, 2*x)*BesselI(1, 2*x)/x-2*BesselI(1, 2*x)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 04 2005
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MAPLE
| seq((binomial(2*n, n)/(1+n))^2, n=0..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 18 2007
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MATHEMATICA
| CatalanNumber[Range[0, 30]]^2 (* From Harvey P. Dale, Apr 26 2011 *)
a[ n_] := If[ n == -1, 0, CatalanNumber[ n]^2] (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ (2 EllipticE[ 16 x] - (1 - 16 x) EllipticK[ 16 x] - Pi/2) / ( 2 Pi x), {x, 0, n}] (* Michael Somos, Jul 11 2011 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ HypergeometricPFQ[ {1/2}, {2, 2}, 4 x^2], {x, 0, 2 n}]] (* Michael Somos, Jul 11 2011 *)
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PROG
| (Mupad) combinat::dyckWords::count(n)^2 $ n = 0..18 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2007
(Sage) [catalan_number(i)^2 for i in xrange(0, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
(PARI) a(n)=(binomial(2*n, n)/(n+1))^2 \\ Charles R Greathouse IV, Jul 16, 2011
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CROSSREFS
| Cf. A000108.
Row sums of triangle A008828.
Sequence in context: A036449 A051500 A206179 * A151342 A202827 A065735
Adjacent sequences: A001243 A001244 A001245 * A001247 A001248 A001249
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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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