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 A057977 GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)). 33
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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 Equal to A063549 (see comments in that sequence). - Nathaniel Johnston, Nov 17 2014 a(n) can be computed with ballot numbers without multiplications or divisions, see Maple program. - Peter Luschny, Feb 23 2019 REFERENCES 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.) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Peter Luschny, The lost Catalan numbers FORMULA 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 From R. J. Mathar, Sep 16 2016: (Start) Conjecture: (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. Conjecture: -(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) EXAMPLE This GCD equals A001405(n) for the smaller odd number gcd(C(12,6), C(11,5)) = gcd(924,462) = 462 = C(11,5). MAPLE 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 b := proc(p, q) option remember; local S;    if p = 0 and q = 0 then return 1 fi;    if p < 0 or  p > q then return 0 fi;    S := b(p-2, q) + b(p, q-2);    if type(q, odd) then S := S + b(p-1, q-1) fi;    S end: seq(b(n, n), n=0..35); # Peter Luschny, Feb 23 2019 MATHEMATICA 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 *) PROG (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 (Sage) def A057977():     x, n = 1, 1     while true:         yield x         m = n if is_odd(n) else 4/(n+2)         x *= m         n += 1 a = A057977(); [a.next() for i in range(36)]   # Peter Luschny, Oct 21 2013 CROSSREFS Cf. A001405, A063549. Bisections are A000108 and A001700. Sequence in context: A184174 A277821 A318280 * A063549 A071653 A227631 Adjacent sequences:  A057974 A057975 A057976 * A057978 A057979 A057980 KEYWORD nonn,easy AUTHOR Labos Elemer, Nov 13 2000 STATUS approved

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Last modified June 25 05:41 EDT 2019. Contains 324346 sequences. (Running on oeis4.)