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editing
approved
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 +759853276142928591932*n -101713943817720) *a(n-3)
83620*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)
-15*(n-3)*(n-6) *(23736630942*n^2 +73277266499*n-235582184233)*a(n-4)=0. - R. J. Mathar, Jul 22 2022
approved
editing
editing
approved
Conjecture D-finite with recurrence -2*(n-1)*(2*n-1) *(7912210314*n^2+24034951267*n-109031255382)*a(n) +2*(-15824420628*n^4+7598532
83620*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)
-15*(n-3)*(n-6) *(23736630942*n^2+73277266499*n-235582184233)*a(n-4)=0. - R. J. Mathar, Jul 22 2022
A045740 := proc(n)
local k ;
add(k*binomial(n, k-1)*binomial(3*n-2*k-1, n-k)/(2*n-k) , k=1..n) ;
end proc:
seq(A045740(n), n=1..30) ; # R. J. Mathar, Jul 22 2022
approved
editing
Emeric Deutsch (deutsch(AT)duke.poly.edu)
Sum(k*binomial(n, k-1)*binomial(3*n-2*k-1, n-k)/(2*n-k), k=1..n)
nonn,new
nonn
Components Number of components in all forests on nodes on a circle.
nonn,new
nonn
sumSum(k*binomial(n,k-1)*binomial(3*n-2*k-1,n-k)/(2*n-k),k=1..n)
nonn,new
nonn
Emeric Deutsch (deutsch@magnus(AT)duke.poly.edu)
Components in all forests on nodes on a circle.
1, 3, 12, 62, 370, 2397, 16345, 115376, 834786, 6152285, 45990120, 347673108, 2652283517, 20385035972, 157656007680, 1225743120520, 9572972899946, 75056029550721, 590469939950716, 4659115833115680, 36859770507695688
1,2
sum(k*binomial(n,k-1)*binomial(3*n-2*k-1,n-k)/(2*n-k),k=1..n)
nonn
Emeric Deutsch (deutsch@magnus.poly.edu)
approved