A045739 - OEIS

%I #16 Mar 08 2023 04:51:45

%S 1,9,70,535,4101,31633,245512,1915875,15020545,118231212,933812892,

%T 7397179309,58746824150,467602683135,3729318261224,29795160492299,

%U 238421091129957,1910544426355420,15329353155160880,123138401704273620

%N Number of edges in all noncrossing forests on n nodes on a circle.

%H Andrew Howroyd, <a href="/A045739/b045739.txt">Table of n, a(n) for n = 2..200</a>

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

%F a(n) = Sum_{k=1..n-1} k*A094040(n, k). - _Andrew Howroyd_, Nov 17 2017

%F 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

%F 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

%o (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

%Y Cf. A094040.

%K nonn

%O 2,2

%A _Emeric Deutsch_