A045722 Number of border edges in all noncrossing rooted trees on n nodes.

1, 6, 28, 150, 858, 5096, 31008, 191862, 1201750, 7597590, 48384180, 309939240, 1994981688, 12892738800, 83604224384, 543722433078, 3545056580814, 23164787710610, 151662849838500, 994674967479270, 6533629880128890

Offset: 1

Author

Emeric Deutsch

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1208
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A026004, A006013.

Programs

Mathematica

MapAt[# - 1 &, Array[(# + 1) Binomial[3 # - 2, # - 1]/# &, 21], 1] (* Michael De Vlieger, Mar 19 2021 *)

Formula

a(n) = (n+1)*binomial(3n-2, n-1)/n for n >= 2. [Corrected by Sean A. Irvine, Mar 19 2021]

G.f.: (1+g-7*g^2+3*g^3)/((1-3*g)*(g-1)^2) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

D-finite with recurrence 2*n*(2*n-1)*a(n) + (-43*n^2+83*n-34)*a(n-1) + 12*(3*n-5)*(3*n-7)*a(n-2) = 0. - R. J. Mathar, Jul 26 2022

a(n) = (n+1) * A006013(n-1) for n >= 2. - F. Chapoton, Feb 27 2024