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Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.
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%I #23 Feb 25 2021 02:24:51

%S 1,1,0,1,2,1,1,5,6,3,1,9,20,22,12,1,14,50,85,91,52,1,20,105,254,385,

%T 408,241,1,27,196,644,1287,1836,1938,1173,1,35,336,1448,3696,6630,

%U 9120,9614,5929,1,44,540,2967,9468,20790,34846,46805,49335,30880

%N Square array read by upwards antidiagonals: T(m,n) is the number of simple 3-connected triangulations of a closed region in the plane with m+3 given external edges and 3n+m internal edges, m>=0, n>=1.

%C A triangulation is simple if it contains no separating 3-cycle. There are n interior nodes and m+3 nodes on the boundary. - _Andrew Howroyd_, Feb 24 2021

%H Andrew Howroyd, <a href="/A210664/b210664.txt">Table of n, a(n) for n = 0..1325</a>

%H P. N. Rathie, <a href="http://dx.doi.org/10.1016/0095-8956(74)90055-0">A census of simple planar triangulations</a>, J. Combin. Theory, B 16 (1974), 134-138.

%H William T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1962-002-9">A census of planar triangulations</a>, Canad. J. Math. 14 (1962), 21-38.

%F From _Andrew Howroyd_, Feb 24 2021: (Start)

%F G.f. of row m > 0: R(x) satisfies g(x^2)^(m+1)*R(x*g(x^2)) = B(x^2) where g(x) is the g.f. of column 0 of A341856 and B(x) is the g.f. of column m of A341856.

%F G.f. of row m > 0: h(x)^(m+1)*B(x*h(x)^2) where 2-h(x) is the g.f. of A000256 and B(x) is the g.f. of column m of A341856.

%F (End)

%e Array begins:

%e 1, 0, 1, 3, 12, ... (A000256)

%e 1, 2, 6, 22, 91, ...

%e 1, 5, 20, 85, 385, ...

%e 1, 9, 50, 254, 1287, ...

%e 1, 14, 105, 644, 3696, ...

%e 1, 20, 196, 1448, 9468, ...

%e ...

%e From _Andrew Howroyd_, Feb 24 2021: (Start)

%e The array transposed for comparability with A341856 begins:

%e ==================================================

%e n\m | 0 1 2 3 4 5 6

%e ----+---------------------------------------------

%e 1 | 1 1 1 1 1 1 1 ...

%e 2 | 0 2 5 9 14 20 27 ...

%e 3 | 1 6 20 50 105 196 336 ...

%e 4 | 3 22 85 254 644 1448 2967 ...

%e 5 | 12 91 385 1287 3696 9468 22131 ...

%e 6 | 52 408 1836 6630 20790 58564 151146 ...

%e 7 | 241 1938 9120 34846 116641 353056 983664 ...

%e (End)

%o (PARI) \\ here H is A000256 as g.f., U(n,m) is A341856 for m > 0.

%o H(n)={my(g=1+serreverse(x/(1+x)^4 + O(x*x^n) )); 2 - sqrt(serreverse(x*(2-g)^2*g^4)/x)}

%o U(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)!))}

%o R(N, m)={my(g=2-H(N)); Vec(if(m==0, 1-g, g^(m+1)*subst(O(x*x^N) + sum(n=1, N, U(n,m)*x^n), x, x*g^2)))}

%o M(m, n=m)={Mat(vectorv(m+1, i, R(n,i-1)))}

%o M(7) \\ _Andrew Howroyd_, Feb 23 2021

%Y Rows m=0..3 are A000256, A000139, A341920, A341921.

%Y Columns are A000012, A000096, A002415, A004305.

%Y Antidiagonal sums give A341922.

%Y Cf. A341856.

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_, Mar 28 2012

%E Terms a(21) and beyond from _Andrew Howroyd_, Feb 23 2021