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%I #20 Jul 26 2022 12:51:11
%S 0,1,13,141,1456,14778,149031,1499773,15089932,151927854,1531242362,
%T 15451614738,156114597744,1579223536788,15993825704427,
%U 162159485143581,1645827425223220,16720488433727910,170023231905932790
%N Number of interior faces in all noncrossing connected graphs on n nodes on a circle.
%H Andrew Howroyd, <a href="/A045742/b045742.txt">Table of n, a(n) for n = 2..200</a>
%F a(n) = Sum_{k=1..n-2} k*binomial(n+k-2, k)*binomial(3*n-3, n-2-k)/(n-1).
%F a(n) = Sum_{k=1..n-2} k*A089434(n,k). - _Andrew Howroyd_, Nov 12 2017
%F a(n) ~ (9 - 5*sqrt(3)) * 2^(n - 5/2) * 3^((3*n-4)/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Dec 10 2021
%F Conjecture D-finite with recurrence n*(n-1)*(523*n-1993)*a(n) -6*(n-1)*(759*n^2-6019*n+12790)*a(n-1) -12*(n-2)*(4707*n^2-17937*n+6260)*a(n-2) +72*(3*n-10)*(759*n-2224)*(3*n-11)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%p A045742 := proc(n)
%p binomial(3*n-3,n-3)*hypergeom([n, 3 - n], [2*n + 1], -1) ;
%p simplify(%) ;
%p end proc: # _R. J. Mathar_, Mar 27 2012
%t Rest[Table[Sum[(k Binomial[n+k-2,k]Binomial[3n-3,n-2-k])/(n-1),{k,n-2}], {n,20}]] (* _Harvey P. Dale_, Nov 29 2011 *)
%o (PARI) a(n) = if(n>1, sum(k=1, n-2, k*binomial(n+k-2, k)*binomial(3*n-3, n-2-k))/(n-1)); \\ _Andrew Howroyd_, Nov 12 2017
%Y Cf. A089434.
%K nonn
%O 2,3
%A _Emeric Deutsch_
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