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%I #45 Apr 24 2020 00:59:27
%S 2,1,2,8,52,495,6470,111034,2419928,65269092,2133844440,83133090480,
%T 3805035352536,202147745618247,12336516593999598,857054350280418290,
%U 67247553674224203280,5917723267088513913032,580407202154922558537264,63093021853191022229671056,7563270705677373923076693840
%N a(n) = 2A(n)/C(n) where A(n) = A180874(n) and C(n) = Catalan(n) = A000108(n).
%C For properties of these numbers including a recurrence, see the Lassalle reference.
%H Andrew Howroyd, <a href="/A188664/b188664.txt">Table of n, a(n) for n = 1..100</a>
%H Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, <a href="https://arxiv.org/abs/1202.1203">A probabilistic interpretation of a sequence related to Narayana Polynomials</a>, arXiv:1202.1203 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 19 2012
%H Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, <a href="https://web.math.rochester.edu/misc/ojac/vol8/68.pdf">A probabilistic interpretation of a sequence related to Narayana Polynomials</a>, Online Journal of Analytic Combinatorics, Issue 8, 2013.
%H Michel Lassalle, <a href="http://arxiv.org/abs/1009.4225">Catalan numbers and a new integer sequence</a>, arXiv:1009.4225 [math.CO], 2010-2012.
%H Michel Lassalle, <a href="https://doi.org/10.1016/j.jcta.2012.01.002">Two integer sequences related to Catalan numbers</a>, Journal of Combinatorial Theory, Series A, Volume 119, Issue 4, May 2012, Pages 923-935.
%p A000108 := proc(n) binomial(2*n,n)/(1+n) ;end proc:
%p 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:
%p A188664 := proc(n) return 2*A180874(n)/A000108(n) ; end proc: # _R. J. Mathar_, Apr 16 2011
%t c = CatalanNumber;
%t 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}]);
%t a[n_] := 2 A[n]/c[n];
%t Array[a, 21] (* _Jean-François Alcover_, Nov 25 2017 *)
%Y Cf. A000108, A180874.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Apr 16 2011
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