A045739 Number of edges in all noncrossing forests on n nodes on a circle.
1, 9, 70, 535, 4101, 31633, 245512, 1915875, 15020545, 118231212, 933812892, 7397179309, 58746824150, 467602683135, 3729318261224, 29795160492299, 238421091129957, 1910544426355420, 15329353155160880, 123138401704273620
Offset: 2
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 2..200
Crossrefs
Cf. A094040.
Programs
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PARI
a(n) = sum(k=1, n-1, k*binomial(n, k+1)*binomial(n+2*k-1, k)/(n+k)); \\ Andrew Howroyd, Nov 12 2017
Formula
a(n) = Sum_{k=1..n-1} k*binomial(n, k+1)*binomial(n+2*k-1, k)/(n+k).
a(n) = Sum_{k=1..n-1} k*A094040(n, k). - Andrew Howroyd, Nov 17 2017
Conjecture D-finite with recurrence -8*(n-1)*(42011*n-237357)*(2*n-1)*a(n) -2*(2*n-3)*(168044*n^2+2298095*n-1588326)*a(n-1) +2*(-8750238*n^3+268256261*n^2-1419101561*n+1999934970)*a(n-2) +4*(137297737*n^3-1755498200*n^2+7473395243*n-10564858105)*a(n-3) +5*(25384204*n^3-350504439*n^2+1587560537*n-2286713022)*a(n-4) -25*(n-4)*(n-7)*(3362075*n-9824604)*a(n-5)=0. - R. J. Mathar, Jul 26 2022
a(n) ~ sqrt(-1 + sqrt(115/111)*sin((Pi + arctan(411*sqrt(111)/2363))/3)) * ((8/3 + 2*sqrt(70)*(sin((Pi + arctan(4537/(111*sqrt(111))))/3)/3))^n / sqrt(Pi*n)). - Vaclav Kotesovec, Mar 08 2023