A341856 Array read by antidiagonals: T(n,k) is the number of rooted strong triangulations of a disk with n interior nodes and 3+k nodes on the boundary.
1, 0, 1, 0, 1, 3, 0, 1, 6, 13, 0, 1, 10, 36, 68, 0, 1, 15, 80, 228, 399, 0, 1, 21, 155, 610, 1518, 2530, 0, 1, 28, 273, 1410, 4625, 10530, 16965, 0, 1, 36, 448, 2933, 12165, 35322, 75516, 118668, 0, 1, 45, 696, 5628, 28707, 102548, 272800, 556512, 857956
Offset: 0
Examples
Array begins: ======================================================= n\k | 0 1 2 3 4 5 6 ----+-------------------------------------------------- 0 | 1 0 0 0 0 0 0 ... 1 | 1 1 1 1 1 1 1 ... 2 | 3 6 10 15 21 28 36 ... 3 | 13 36 80 155 273 448 696 ... 4 | 68 228 610 1410 2933 5628 10128 ... 5 | 399 1518 4625 12165 28707 62230 125928 ... 6 | 2530 10530 35322 102548 267162 638624 1422204 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325
- William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.
Crossrefs
Programs
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PARI
T(n,m)=if(m==0, 2*(4*n+1)!/((3*n+2)!*(n+1)!), (3*(m+2)!*(m-1)!/(3*n+3*m+3)!)*sum(j=0, min(m,n-1), (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!)))
Formula
T(n,0) = A000260(n) = 2*(4*n+1)!/((3*n+2)!*(n+1)!).
T(n,m) = (3*(m+2)!*(m-1)!/(3*n+3*m+3)!) * Sum_{j=0..min(m,n-1)} (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!) for m > 0.
Comments