A276666 a(n) = (n-1)*Catalan(n).

-1, 0, 2, 10, 42, 168, 660, 2574, 10010, 38896, 151164, 587860, 2288132, 8914800, 34767720, 135727830, 530365050, 2074316640, 8119857900, 31810737420, 124718287980, 489325340400, 1921133836440, 7547311500300, 29667795388452, 116686713634848, 459183826803800

Offset: 0

Author

Peter Luschny, Sep 12 2016

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

A024483 is a variant of this sequence.

Cf. A000108, A000984, A001622, A051631, A056040, A057977, A232500.

Programs

GAP

Concatenation([-1], List([1..30], n-> 2*Binomial(2*n-1, n+1))); # G. C. Greubel, Aug 29 2019

Magma

[(n-1)*Catalan(n): n in [0..30]]; // Vincenzo Librandi, Sep 13 2016

Maple

f := (1-3*x)/(x*sqrt(1-4*x))-1/x:
series(f,x,29): seq(coeff(%,x,n), n=0..26);
A276666 := n -> (n^2-1)*(2*n)!/(n+1)!^2:
seq(A276666(n), n=0..26);

Mathematica

Table[(n - 1) CatalanNumber[n], {n, 0, 30}] (* Vincenzo Librandi, Sep 13 2016 *)

PARI

a(n) = if(n==0,-1, 2*binomial(2*n-1, n+1)); \\ G. C. Greubel, Aug 29 2019

Sage

A276666 = lambda n: (n - 1) * catalan_number(n)
[A276666(n) for n in range(27)]

Formula

a(n) = [x^n] (1-3*x)/(x*sqrt(1-4*x))-1/x.

a(n) = 4^n*(n-1)*hypergeom([3/2, -n], [2], 1).

a(n) = 4^n*(n-1)*JacobiP(n,1,-1/2-n,-1)/(n+1).

a(n) = (2*n)! [x^(2^n)]( BesselI(2,2*x) - (1+1/x)*BesselI(1,2*x) ).

a(n) = binomial(2*n,n) - 2*Catalan(n). (See Geoffrey Critzer's formula in A024483).

a(n) = A056040(2*n) - 2*A057977(2*n).

a(n) = A056040(2*n)*(1-2/(n+1)) = (n^2-1)*(2*n)!/(n+1)!^2.

a(n) = A232500(2*n).

a(n) = a(n-1)*2*(n-1)*(2*n-1)/((n-2)*(n+1)) for n > 2. - Chai Wah Wu, Sep 12 2016

a(n) = A024483(n+1) for n>0. - R. J. Mathar, Sep 13 2016

a(n) = A000984(n+1)-3*A000984(n). - Ezhilarasu Velayutham, Aug 27 2019

From Amiram Eldar, Mar 22 2022: (Start)

Sum_{n>=2} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).

Sum_{n>=2} (-1)^n/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)