%I #59 Aug 20 2022 09:00:16
%S 1,1,1,3,2,10,5,35,14,126,42,462,132,1716,429,6435,1430,24310,4862,
%T 92378,16796,352716,58786,1352078,208012,5200300,742900,20058300,
%U 2674440,77558760,9694845,300540195,35357670,1166803110,129644790,4537567650
%N GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)).
%C 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
%C 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
%C Equal to A063549 (see comments in that sequence). - _Nathaniel Johnston_, Nov 17 2014
%C a(n) can be computed with ballot numbers without multiplications or divisions, see Maple program. - _Peter Luschny_, Feb 23 2019
%H Vincenzo Librandi, <a href="/A057977/b057977.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.
%F 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
%F From _Peter Luschny_, Apr 30 2011: (Start)
%F Recurrence: a(0) = 1 and a(n) = a(n-1)*n^[n odd]*(4/(n+2))^[n even] for n > 0.
%F 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))).
%F 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)
%F 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
%F From _R. J. Mathar_, Sep 16 2016: (Start)
%F D-finite with recurrence: (n+2)*a(n) - n*a(n-1) + 4*(-2*n+1)*a(n-2) + 4*(n-1)*a(n-3) + 16*(n-3)*a(n-4) = 0.
%F D-finite with recurrence: -(n+2)*(n^2-5)*a(n) + 4*(-2*n-1)*a(n-1) + 4*(n-1)*(n^2+2*n-4)*a(n-2) = 0. (End)
%F Sum_{n>=0} 1/a(n) = 8/3 + 8*Pi/(9*sqrt(3)). - _Amiram Eldar_, Aug 20 2022
%e This GCD equals A001405(n) for the smaller odd number gcd(C(12,6), C(11,5)) = gcd(924,462) = 462 = C(11,5).
%p A057977_ogf := proc(z) b := z -> (z-1)/(2*z^2);
%p (2 + b(z))/sqrt(1-4*z^2) - b(z) end:
%p seq(coeff(series(A057977_ogf(z),z,n+3),z,n), n = 0..35);
%p A057977_rec := n -> `if`(n=0, 1, A057977_rec(n-1)*n^modp(n,2)
%p *(4/(n+2))^modp(n+1,2));
%p 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:
%p A057977 := n -> (n!/iquo(n,2)!^2) / (iquo(n,2)+1):
%p seq(A057977(n), n=0..35); # _Peter Luschny_, Apr 30 2011
%p b := proc(p, q) option remember; local S;
%p if p = 0 and q = 0 then return 1 fi;
%p if p < 0 or p > q then return 0 fi;
%p S := b(p-2, q) + b(p, q-2);
%p if type(q, odd) then S := S + b(p-1, q-1) fi;
%p S end:
%p seq(b(n, n), n=0..35); # _Peter Luschny_, Feb 23 2019
%t 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_ *)
%o (PARI) a(n)=if(n<0,0,(n+n%2)!/(n\2+1)!/(n\2+n%2)!/(1+n%2))
%o a(n)=n!/(n\2)!^2/(n\2+1) \\ _Charles R Greathouse IV_, May 02, 2011
%o (Sage)
%o def A057977():
%o x, n = 1, 1
%o while True:
%o yield x
%o m = n if is_odd(n) else 4/(n+2)
%o x *= m
%o n += 1
%o a = A057977(); [next(a) for i in range(36)] # _Peter Luschny_, Oct 21 2013
%Y Cf. A001405, A063549.
%Y Bisections are A000108 and A001700.
%K nonn,easy
%O 0,4
%A _Labos Elemer_, Nov 13 2000