A188664 a(n) = 2A(n)/C(n) where A(n) = A180874(n) and C(n) = Catalan(n) = A000108(n).

2, 1, 2, 8, 52, 495, 6470, 111034, 2419928, 65269092, 2133844440, 83133090480, 3805035352536, 202147745618247, 12336516593999598, 857054350280418290, 67247553674224203280, 5917723267088513913032, 580407202154922558537264, 63093021853191022229671056, 7563270705677373923076693840

Offset: 1

Author

N. J. A. Sloane, Apr 16 2011

Keywords

• nonn

Comments

For properties of these numbers including a recurrence, see the Lassalle reference.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100
• Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, A probabilistic interpretation of a sequence related to Narayana Polynomials, arXiv:1202.1203 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 19 2012
• Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, A probabilistic interpretation of a sequence related to Narayana Polynomials, Online Journal of Analytic Combinatorics, Issue 8, 2013.
• Michel Lassalle, Catalan numbers and a new integer sequence, arXiv:1009.4225 [math.CO], 2010-2012.
• Michel Lassalle, Two integer sequences related to Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 119, Issue 4, May 2012, Pages 923-935.

Crossrefs

Cf. A000108, A180874.

Programs

Maple

A000108 := proc(n) binomial(2*n,n)/(1+n) ;end proc:
A180874 := proc(n) option remember; if n = 1 then 1 else A000108(n)+add((-1)^j*binomial(2*n-1,2*j-1)*procname(j)*A000108(n-j),j=1..n-1) ;  %*(-1)^(n-1) ; end if; end proc:
A188664 := proc(n) return 2*A180874(n)/A000108(n) ; end proc: # R. J. Mathar, Apr 16 2011

Mathematica

c = CatalanNumber;
A[n_] := A[n] = (-1)^(n - 1)*(c[n] + Sum[(-1)^j*Binomial[2*n - 1, 2*j - 1]*A[j]*c[n - j], {j, 1, n - 1}]);
a[n_] := 2 A[n]/c[n];
Array[a, 21] (* Jean-François Alcover, Nov 25 2017 *)