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 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. 9
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS A strong triangulation is one in which no interior edge joins two nodes on the boundary. Except for the single triangle which is enumerated by T(0,0) these are the 3-connected triangulations. 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. 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. EXAMPLE 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 ... ... PROG (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)!))) CROSSREFS Columns k=0..3 are A000260, A242136, A341917, A341918. Antidiagonal sums give A341919. Cf. A146305 (not necessarily strong triangulations), A210664, A341923, A342053. Sequence in context: A243984 A100485 A143397 * A339350 A244118 A273155 Adjacent sequences: A341853 A341854 A341855 * A341857 A341858 A341859 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Feb 23 2021 STATUS approved

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)