A146305 Array T(n,m) = 2*(2m+3)!*(4n+2m+1)!/(m!*(m+2)!*n!*(3n+2m+3)!) read by antidiagonals.

1, 1, 2, 3, 5, 5, 13, 20, 21, 14, 68, 100, 105, 84, 42, 399, 570, 595, 504, 330, 132, 2530, 3542, 3675, 3192, 2310, 1287, 429, 16965, 23400, 24150, 21252, 16170, 10296, 5005, 1430, 118668, 161820, 166257, 147420, 115500, 78936, 45045, 19448, 4862, 857956

Offset: 0

Author

R. J. Mathar, Oct 29 2008

Keywords

• easy
• nonn
• tabl

Comments

T(n,m) is the number of rooted nonseparable (2-connected) triangulations of the disk with n internal nodes and 3 + m nodes on the external face. The triangulation has 2*n + m + 1 triangles and 3*(n+1) + 2*m edges. - Andrew Howroyd, Feb 21 2021

Examples

			The array starts at row n=0 and column m=0 as
.....1......2.......5......14.......42.......132
.....1......5......21......84......330......1287
.....3.....20.....105.....504.....2310.....10296
....13....100.....595....3192....16170.....78936
....68....570....3675...21252...115500....602316
...399...3542...24150..147420...844074...4628052
..2530..23400..166257.1057224..6301680..35939904
.16965.161820.1186680.7791168.47948670.282285432
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325
• Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
• William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

Crossrefs

Columns m=0..3 are A000260, A197271(n+1), A341853, A341854.

Rows n=0..2 are A000108(n+1), A002054(n+1) and A000917.

Antidiagonal sums are A000260(n+1).

Cf. A169808 (unrooted), A169809 (achiral), A262586 (oriented).

Programs

Maple

A146305 := proc(n,m)
    2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ;
end proc:
for d from 0 to 13 do
    for m from 0 to d do
        printf("%d,", A146305(d-m,m)) ;
    end do:
end do:

Mathematica

T[n_, m_] := 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)!; Table[T[n-m, m], {n, 0, 13}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 06 2014, after Maple *)

PARI

T(n,m)={2*(2*m+3)!*(4*n+2*m+1)!/(m!*(m+2)!*n!*(3*n+2*m+3)!)} \\ Andrew Howroyd, Feb 21 2021