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A057977 GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)).

%I A057977 #61 Jun 29 2025 21:31:14
%S A057977 1,1,1,3,2,10,5,35,14,126,42,462,132,1716,429,6435,1430,24310,4862,
%T A057977 92378,16796,352716,58786,1352078,208012,5200300,742900,20058300,
%U A057977 2674440,77558760,9694845,300540195,35357670,1166803110,129644790,4537567650
%N A057977 GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)).
%C A057977 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 A057977 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 A057977 Equal to A063549 (see comments in that sequence). - _Nathaniel Johnston_, Nov 17 2014
%C A057977 a(n) can be computed with ballot numbers without multiplications or divisions, see Maple program. - _Peter Luschny_, Feb 23 2019
%H A057977 Vincenzo Librandi, <a href="/A057977/b057977.txt">Table of n, a(n) for n = 0..1000</a>
%H A057977 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A057977 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.
%F A057977 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 A057977 From _Peter Luschny_, Apr 30 2011: (Start)
%F A057977 Recurrence: a(0) = 1 and a(n) = a(n-1)*n^[n odd]*(4/(n+2))^[n even] for n > 0.
%F A057977 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 A057977 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 A057977 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 A057977 From _R. J. Mathar_, Sep 16 2016: (Start)
%F A057977 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 A057977 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 A057977 Sum_{n>=0} 1/a(n) = 8/3 + 8*Pi/(9*sqrt(3)). - _Amiram Eldar_, Aug 20 2022
%e A057977 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 A057977_ogf := proc(z) b := z -> (z-1)/(2*z^2);
%p A057977 (2 + b(z))/sqrt(1-4*z^2) - b(z) end:
%p A057977 seq(coeff(series(A057977_ogf(z),z,n+3),z,n), n = 0..35);
%p A057977 A057977_rec := n -> `if`(n=0, 1, A057977_rec(n-1)*n^modp(n,2)
%p A057977 *(4/(n+2))^modp(n+1,2));
%p A057977 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 A057977 := n -> (n!/iquo(n,2)!^2) / (iquo(n,2)+1):
%p A057977 seq(A057977(n), n=0..35); # _Peter Luschny_, Apr 30 2011
%p A057977 b := proc(p, q) option remember; local S;
%p A057977    if p = 0 and q = 0 then return 1 fi;
%p A057977    if p < 0 or  p > q then return 0 fi;
%p A057977    S := b(p-2, q) + b(p, q-2);
%p A057977    if type(q, odd) then S := S + b(p-1, q-1) fi;
%p A057977    S end:
%p A057977 seq(b(n, n), n=0..35); # _Peter Luschny_, Feb 23 2019
%t A057977 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 A057977 (PARI) a(n)=if(n<0,0,(n+n%2)!/(n\2+1)!/(n\2+n%2)!/(1+n%2))
%o A057977 a(n)=n!/(n\2)!^2/(n\2+1) \\ _Charles R Greathouse IV_, May 02 2011
%o A057977 (Sage)
%o A057977 def A057977():
%o A057977     x, n = 1, 1
%o A057977     while True:
%o A057977         yield x
%o A057977         m = n if is_odd(n) else 4/(n+2)
%o A057977         x *= m
%o A057977         n += 1
%o A057977 a = A057977(); [next(a) for i in range(36)]   # _Peter Luschny_, Oct 21 2013
%Y A057977 Cf. A001405, A063549.
%Y A057977 Bisections are A000108 and A001700.
%K A057977 nonn,easy
%O A057977 0,4
%A A057977 _Labos Elemer_, Nov 13 2000