A045742 Number of interior faces in all noncrossing connected graphs on n nodes on a circle.

0, 1, 13, 141, 1456, 14778, 149031, 1499773, 15089932, 151927854, 1531242362, 15451614738, 156114597744, 1579223536788, 15993825704427, 162159485143581, 1645827425223220, 16720488433727910, 170023231905932790

Offset

2,3

Links

Andrew Howroyd, Table of n, a(n) for n = 2..200

Formula

a(n) = Sum_{k=1..n-2} k*binomial(n+k-2, k)*binomial(3*n-3, n-2-k)/(n-1).

a(n) = Sum_{k=1..n-2} k*A089434(n,k). - Andrew Howroyd, Nov 12 2017

a(n) ~ (9 - 5*sqrt(3)) * 2^(n - 5/2) * 3^((3*n-4)/2) / sqrt(Pi*n). - Vaclav Kotesovec, Dec 10 2021

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

Maple

A045742 := proc(n)
        binomial(3*n-3, n-3)*hypergeom([n, 3 - n], [2*n + 1], -1) ;
        simplify(%) ;
end proc: # R. J. Mathar, Mar 27 2012

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A089434.

Sequence in context: A157160 A263480 A210766 * A122011 A221103 A239250

Adjacent sequences: A045739 A045740 A045741 * A045743 A045744 A045745

Keyword

nonn

Author

Emeric Deutsch

Status

approved