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A057977
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GCD of consecutive central binomial coefficients: a(n) = GCD[A001405(n+1), A001405(n)].
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10
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1, 1, 1, 3, 2, 10, 5, 35, 14, 126, 42, 462, 132, 1716, 429, 6435, 1430, 24310, 4862, 92378, 16796, 352716, 58786, 1352078, 208012, 5200300, 742900, 20058300, 2674440, 77558760, 9694845, 300540195, 35357670, 1166803110, 129644790, 4537567650
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OFFSET
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0,4
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COMMENTS
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The numbers can be seen as a generalization of the Catalan numbers, extending A000984(n)/(n+1) to A056040(n)/(floor(n/2)+1). They can also be seen as composing the aerated Catalan numbers A126120 with the aerated complementary Catalan numbers A138364. (Thus the name 'extended Catalan numbers' might be apt for this sequence.) - Peter Luschny, May 03 2011
a(n) is the number of lattice paths from (0,0) to (n,0) that do not go below the x-axis and consist of steps U=(1,1), D=(1,-1) and maximally one step H=(1,0). - Alois P. Heinz, Apr 17 2013
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REFERENCES
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Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008. (Includes 14 hypergeometric representations of a(n) and a combinatorial interpretation of a(n) as the number of orbital systems.)
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Luschny, The lost Catalan numbers
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FORMULA
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G.f.: (4*x^2+x-1+(1-x)*sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*x^2). E.g.f.: (1+1/x)*BesselI(1, 2*x). - Vladeta Jovovic, Jan 19 2004
From Peter Luschny, Apr 30 2011: (Start)
Recurrence: a(0) = 1 and a(n) = a(n-1)*n^[n odd]*(4/(n+2))^[n even] for n > 0.
Asymptotic formula: Let [n even] = 1 if n is even, 0 otherwise. Let N := n+1+[n even]. Then a(n) ~ 2^N /((n+1)^[n even]*sqrt(Pi*(2*N+1))).
Integral representation: a(n) = (1/(2*Pi))*Int_{x=0..4}(x^(2*n-1)* ((4-x)^2/x)^cos(Pi*n))^(1/4) (End)
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+2)//U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
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EXAMPLE
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This GCD equals A001405(n) for the smaller odd number GCD[C[12,6],C[11,5]]=GCD[924,462]=462=C[11,5]
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MAPLE
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A057977_ogf := proc(z) b := z -> (z-1)/(2*z^2);
(2 + b(z))/sqrt(1-4*z^2) - b(z) end:
seq(coeff(series(A057977_ogf(z), z, n+3), z, n), n = 0..35);
A057977_rec := n -> `if`(n=0, 1, A057977_rec(n-1)*n^modp(n, 2)
*(4/(n+2))^modp(n+1, 2));
A057977_int := proc(n) int((x^(2*n-1)*((4-x)^2/x)^cos(Pi*n))^(1/4), x=0..4)/(2*Pi); round(evalf(%)) end:
A057977 := n -> (n!/iquo(n, 2)!^2) / (iquo(n, 2)+1):
seq(A057977(n), n=0..35); # Peter Luschny, Apr 30 2011
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MATHEMATICA
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a[n_] := n! / (Quotient[n, 2]!^2 * (Quotient[n, 2]+1)); Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 03 2012, after Peter Luschny *)
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PROG
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(PARI) a(n)=if(n<0, 0, (n+n%2)!/(n\2+1)!/(n\2+n%2)!/(1+n%2))
a(n)=n!/(n\2)!^2/(n\2+1) \\ Charles R Greathouse IV, May 02, 2011
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CROSSREFS
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Cf. A001405.
Bisections are A000108 and A001700.
Sequence in context: A090780 A184174 * A063549 A071653 A056861 A214844
Adjacent sequences: A057974 A057975 A057976 * A057978 A057979 A057980
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KEYWORD
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nonn,easy
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Nov 13 2000
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STATUS
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approved
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