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%I #18 Mar 22 2023 17:34:37
%S 1,1,2,3,5,5,13,20,21,14,68,100,105,84,42,399,570,595,504,330,132,
%T 2530,3542,3675,3192,2310,1287,429,16965,23400,24150,21252,16170,
%U 10296,5005,1430,118668,161820,166257,147420,115500,78936,45045,19448,4862,857956
%N Array T(n,m) = 2*(2m+3)!*(4n+2m+1)!/(m!*(m+2)!*n!*(3n+2m+3)!) read by antidiagonals.
%C 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
%H Andrew Howroyd, <a href="/A146305/b146305.txt">Table of n, a(n) for n = 0..1325</a>
%H Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, <a href="https://arxiv.org/abs/2303.10986">Refined product formulas for Tamari intervals</a>, arXiv:2303.10986 [math.CO], 2023.
%H William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
%e The array starts at row n=0 and column m=0 as
%e .....1......2.......5......14.......42.......132
%e .....1......5......21......84......330......1287
%e .....3.....20.....105.....504.....2310.....10296
%e ....13....100.....595....3192....16170.....78936
%e ....68....570....3675...21252...115500....602316
%e ...399...3542...24150..147420...844074...4628052
%e ..2530..23400..166257.1057224..6301680..35939904
%e .16965.161820.1186680.7791168.47948670.282285432
%p A146305 := proc(n,m)
%p 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ;
%p end proc:
%p for d from 0 to 13 do
%p for m from 0 to d do
%p printf("%d,", A146305(d-m,m)) ;
%p end do:
%p end do:
%t 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 *)
%o (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
%Y Columns m=0..3 are A000260, A197271(n+1), A341853, A341854.
%Y Rows n=0..2 are A000108(n+1), A002054(n+1) and A000917.
%Y Antidiagonal sums are A000260(n+1).
%Y Cf. A169808 (unrooted), A169809 (achiral), A262586 (oriented).
%K easy,nonn,tabl
%O 0,3
%A _R. J. Mathar_, Oct 29 2008
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