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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, 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
0,6
LINKS
FORMULA
G.f.: 1/(1 - z - tz^2 - tz^3 - t^2z^4).
EXAMPLE
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;
...
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
PROG
(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
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).
Sequence in context: A130323 A130303 A100898 * A199478 A134867 A102573
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 25 2004
STATUS
approved