A049607 Duplicate of A035029.
0, 1, 5, 26, 138, 743, 4043, 22180, 122468, 679757, 3789297, 21199998, 118973550
Offset: 0
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
I:=[4,20,104]; [n le 3 select I[n] else ( (7*n+1)*Self(n-1) - (7*n-15)*Self(n-2) + (n-3)*Self(n-3) )/(n+1): n in [1..30]]; // G. C. Greubel, Oct 19 2022
Differences[CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x,0,30}], x]] (* Harvey P. Dale, Jun 04 2011 *)
With[{P=LegendreP}, Table[(n*(n+2)*P[n+2,3] -(6*n^2+10*n+1)*P[n+1,3] +(n+1)*(5*n+ 3)*P[n,3])/(2*n*(n+1)), {n,30}]] (* G. C. Greubel, Oct 19 2022 *)
def A001850(n): return gen_legendre_P(n,0,3) def A035028(n): return ((n+1)*(n+3)*A001850(n+3) - (6*n^2 +22*n +17)*A001850(n+2) + (n+2)*(5*n+8)*A001850(n+1))/(2*(n+1)*(n+2)) [A035028(n) for n in range(40)] # G. C. Greubel, Oct 19 2022
A convex 3-gon is a triangle. There is only one dissection of a rooted triangle, with one single part. The factorial moment of order two is therefore 0 and hence a(1) = 0. A convex 4-gon is a quadrilateral. There are three dissections of a rooted quadrilateral, two with two parts and one with one part. Then the expectation of the number of parts is 5/3, and the expectation of the number of parts squared is 9/3, hence the factorial moment of order two is 9/3 - 5/3 = 4/3. The second Schröder number is A001003(2) = 3, therefore a(2) = 4.
s := (z^2-6*z+1)^(1/2): g := z/s^3-(1/s-(z+1-s)/(4*z))/2: ser := series(g,z,30): seq(coeff(ser,z,n), n=0..23); # Peter Luschny, Nov 17 2016
CoefficientList[Series[z/Sqrt[(z^2 - 6*z + 1)^3] - (1/Sqrt[z^2 - 6*z + 1] - (z + 1 - Sqrt[z^2 - 6*z + 1])/(4*z))/2, {z, 0, 20}], z]
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