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Number of components in all forests on nodes on a circle.
1

%I #9 Jan 11 2024 08:02:43

%S 1,3,12,62,370,2397,16345,115376,834786,6152285,45990120,347673108,

%T 2652283517,20385035972,157656007680,1225743120520,9572972899946,

%U 75056029550721,590469939950716,4659115833115680,36859770507695688

%N Number of components in all forests on nodes on a circle.

%F Sum(k*binomial(n, k-1)*binomial(3*n-2*k-1, n-k)/(2*n-k), k=1..n)

%F Conjecture D-finite with recurrence -2*(n-1)*(2*n-1) *(7912210314*n^2 +24034951267*n -109031255382)*a(n) +2*(-15824420628*n^4 +759853283620*n^3 -1653756416501*n^2 -3170366114943*n +6074871939666) *a(n-1) +2*(1171007126472*n^4 -5580539787848*n^3 -21281457754861*n^2 +151349953543323*n -205322404158756) *a(n-2) +(530118091038*n^4 -3085109917817*n^3 -8408054715093*n^2 +76142928591932*n -101713943817720) *a(n-3)

%F -15*(n-3)*(n-6) *(23736630942*n^2 +73277266499*n-235582184233)*a(n-4)=0. - _R. J. Mathar_, Jul 22 2022

%p A045740 := proc(n)

%p local k ;

%p add(k*binomial(n,k-1)*binomial(3*n-2*k-1,n-k)/(2*n-k) ,k=1..n) ;

%p end proc:

%p seq(A045740(n),n=1..30) ; # _R. J. Mathar_, Jul 22 2022

%K nonn

%O 1,2

%A _Emeric Deutsch_