A101350 Triangle read by rows: T(n,k) = number of k-matchings in the graph obtained by a zig-zag triangulation of a convex n-gon, T(0,0)=T(1,0)=T(2,0)=T(2,1)=1 (n > 2, 0 <= k <= floor(n/2)).

1, 1, 1, 1, 1, 3, 1, 5, 2, 1, 7, 7, 1, 9, 16, 3, 1, 11, 29, 15, 1, 13, 46, 43, 5, 1, 15, 67, 95, 30, 1, 17, 92, 179, 104, 8, 1, 19, 121, 303, 271, 58, 1, 21, 154, 475, 591, 235, 13, 1, 23, 191, 703, 1140, 705, 109, 1, 25, 232, 995, 2010, 1746, 506, 21, 1, 27, 277, 1359, 3309, 3780

Offset: 0

Author

Emeric Deutsch, Dec 25 2004

Keywords

• nonn
• tabf

Examples

			T(5,2)=7 because in the triangulation of the convex pentagon ABCDEA with diagonals AD and AC we have seven 2-matchings: {AB,CD},{AB,DE},{BC,AD},{BC,DE},{BC,EA},{CD,EA} and {DE,AC}.
Triangle begins:
  1;
  1;
  1,  1;
  1,  3;
  1,  5,  2;
  1,  7,  7;
  1,  9, 16,  3;
  1, 11, 29, 15;
  1, 13, 46, 43, 5;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..649

Crossrefs

Row sums yield A000078 (the tetranacci numbers). T(2n+1, n) = A023610(n) (n > 0). T(2n, n) = A000045(n+1) (the Fibonacci numbers).

Cf. A000078, A023610, A000045.

Programs

Maple

G:=1/(1-z-t*z^2-t*z^3-t^2*z^4):Gserz:=simplify(series(G,z=0,18)):P[0]:=1: for n from 1 to 16 do P[n]:=sort(coeff(Gserz,z^n)) od:for n from 0 to 16 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od;# yields the sequence in triangular form

PARI

s(n) = 1/(1-x-y*x^2-y*x^3-y^2*x^4) + O(x^n);
my(gf=Pol(s(20))); for(n=0, poldegree(gf), my(p=polcoeff(gf,n)); for(k=0, poldegree(p), print1(polcoeff(p,k), ", ")); print) \\ Andrew Howroyd, Nov 04 2017

Formula

G.f.: 1/(1 - z - tz^2 - tz^3 - t^2z^4).