A282735 Number of triangulations of a convex n-gon in the plane each of whose sides is subdivided by 2 points.

2, 29, 604, 13740, 332842, 8419334, 219829560, 5880376980, 160327187560, 4439180561629, 124485616405516, 3528306241927428, 100914357176842188, 2908920668838209340, 84423279647430363248, 2464811859798957024196, 72343319478816485276760, 2133323371103124457168580

Offset

2,1

Links

Lars Blomberg, Table of n, a(n) for n = 2..101

Andrei Asinowski, Christian Krattenthaler, Toufik Mansour, Counting triangulations of some classes of subdivided convex polygons, arXiv:1604.02870 [math.CO], 2016.

Formula

From Asinowski and Krattenthaler equation 2.7: a(n) = tr(n,3). - Lars Blomberg, Mar 04 2017

From Seiichi Manyama, Apr 30 2026: (Start)

a(n) = [x^(3*n-2)] (1-x) * (1+x)^(3*n-4) * (1+x+x^2+x^3)^n.

a(n) = Sum_{i=0..3*n-2} binomial(3*n-4,i) * Sum_{j=0..floor((3*n-i-2)/4)} (-1)^j * binomial(n,j) * binomial(4*n-i-4-4*j,n-2). (End)

Mathematica

tr[k_, r_] := Sum[(-1)^j 2^l Binomial[k, j] Binomial[k-2+l, l] Binomial[(r- 1)k-l-3, r k - (r+1)j-l-2], {j, 0, k}, {l, 0, r k - (r+1)j-2}];
a[n_] := tr[n, 3];
Table[a[n], {n, 2, 19}] (* Jean-François Alcover, Oct 10 2018 *)

Prog

(PARI) a(n, k=3) = sum(i=0, n*k-2, binomial(n*k-4, i)*sum(j=0, (n*k-i-2)\(k+1), (-1)^j*binomial(n, j)*binomial(n*(k+1)-i-4-(k+1)*j, n-2))); \\ Seiichi Manyama, Apr 30 2026

Crossrefs

Columns k=3 of A395602.

Cf. A086452, A282736.

Sequence in context: A187362 A176938 A006988 * A245252 A090251 A087281

Adjacent sequences: A282732 A282733 A282734 * A282736 A282737 A282738

Keyword

nonn

Author

N. J. A. Sloane, Mar 03 2017

Extensions

More terms from Lars Blomberg, Mar 04 2017

Status

approved