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

%I A146305 #18 Mar 22 2023 17:34:37
%S A146305 1,1,2,3,5,5,13,20,21,14,68,100,105,84,42,399,570,595,504,330,132,
%T A146305 2530,3542,3675,3192,2310,1287,429,16965,23400,24150,21252,16170,
%U A146305 10296,5005,1430,118668,161820,166257,147420,115500,78936,45045,19448,4862,857956
%N A146305 Array T(n,m) = 2*(2m+3)!*(4n+2m+1)!/(m!*(m+2)!*n!*(3n+2m+3)!) read by antidiagonals.
%C A146305 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 A146305 Andrew Howroyd, <a href="/A146305/b146305.txt">Table of n, a(n) for n = 0..1325</a>
%H A146305 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 A146305 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 A146305 The array starts at row n=0 and column m=0 as
%e A146305 .....1......2.......5......14.......42.......132
%e A146305 .....1......5......21......84......330......1287
%e A146305 .....3.....20.....105.....504.....2310.....10296
%e A146305 ....13....100.....595....3192....16170.....78936
%e A146305 ....68....570....3675...21252...115500....602316
%e A146305 ...399...3542...24150..147420...844074...4628052
%e A146305 ..2530..23400..166257.1057224..6301680..35939904
%e A146305 .16965.161820.1186680.7791168.47948670.282285432
%p A146305 A146305 := proc(n,m)
%p A146305     2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ;
%p A146305 end proc:
%p A146305 for d from 0 to 13 do
%p A146305     for m from 0 to d do
%p A146305         printf("%d,", A146305(d-m,m)) ;
%p A146305     end do:
%p A146305 end do:
%t A146305 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 A146305 (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 A146305 Columns m=0..3 are A000260, A197271(n+1), A341853, A341854.
%Y A146305 Rows n=0..2 are A000108(n+1), A002054(n+1) and A000917.
%Y A146305 Antidiagonal sums are A000260(n+1).
%Y A146305 Cf. A169808 (unrooted), A169809 (achiral), A262586 (oriented).
%K A146305 easy,nonn,tabl
%O A146305 0,3
%A A146305 _R. J. Mathar_, Oct 29 2008