login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

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

MATHEMATICA

a[n_] := n! / (Quotient[n, 2]!^2 * (Quotient[n, 2]+1)); Table[a[n], {n, 0, 35}] (* Jean-Fran├žois 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: A277821 A318280 * A063549 A071653 A227631 A246830

Adjacent sequences:  A057974 A057975 A057976 * A057978 A057979 A057980

KEYWORD

nonn,easy

AUTHOR

Labos Elemer, Nov 13 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 22 18:55 EDT 2018. Contains 316500 sequences. (Running on oeis4.)