A093127 Triangle read by rows: differences of Narayana numbers.

1, 1, 1, 2, 1, 5, 1, 9, 3, 1, 14, 14, 1, 20, 40, 4, 1, 27, 90, 30, 1, 35, 175, 125, 5, 1, 44, 308, 385, 55, 1, 54, 504, 980, 315, 6, 1, 65, 780, 2184, 1274, 91, 1, 77, 1155, 4410, 4116, 686, 7, 1, 90, 1650, 8250, 11340, 3528, 140, 1, 104, 2288, 14520, 27720, 14112, 1344, 8

Offset: 0

Author

David Callan, Mar 23 2004

Keywords

• easy
• nonn
• tabf

Comments

T(n,k) (0 <= k <= n/2) is the number of dissections of a regular (n+2)-gon using k strictly disjoint diagonals (diagonals join nonconsecutive vertices and strictly disjoint means no two cross or share an endpoint).

Row n contains 1 + floor(n/2) entries. - Emeric Deutsch, Sep 18 2014

T(n,k) is the number of paths in B(n+1) that do not start with H and have k steps of weight 2 (i.e., H or u). Example: T(3,1) = 5 because the paths in B(4) that do not start with H are hhhh, hHh, hhH, uhd, hud, and udh; the last five contain exactly 1 step of weight 2. - Emeric Deutsch, Sep 18 2014

B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps. - Emeric Deutsch, Sep 18 2014

Examples

			T(3,1) = 5 because there are 5 ways to insert a single diagonal into a pentagon.
Triangle begins:
  1;
  1;
  1,  2;
  1,  5;
  1,  9,  3;
  1, 14, 14;
  1, 20, 40,  4;
  1, 27, 90, 30;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306. - _Emeric Deutsch_, Sep 18 2014

Programs

GAP

B:=Binomial;;
T:= function(n,k)
    return B(n-k+2,k+1)*B(n-k+2,k)/(n-k+2) - B(n-k+1,k)*B(n-k+1,k-1)/(n-k+1);
    end;
Flat(List([0..15], n-> List([0..Int(n/2)], k-> T(n,k) ))); # G. C. Greubel, Dec 28 2019

Magma

B:=Binomial; T:= func< n, k | B(n-k+2,k+1)*B(n-k+2,k)/(n-k+2) - B(n-k+1,k)*B(n-k+1,k-1)/(n-k+1) >;
[T(n,k): k in [0..Floor(n/2)], n in [0..15]]; // G. C. Greubel, Dec 28 2019

Maple

eq := t*z^4*G^2-(1-z-2*t*z^2-t*z^3+t^2*z^4)*G+1 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form # Emeric Deutsch, Sep 18 2014
T := proc (n, k) if (1/2)*n+1/2 < k then 0 else binomial(n-k+2, k+1)*binomial(n-k+2, k)/(n-k+2)-binomial(n-k+1, k)*binomial(n-k+1, k-1)/(n-k+1) end if end proc: for n from 0 to 15 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form # Emeric Deutsch, Sep 18 2014

Mathematica

Nara[n_, k_]:= Binomial[n, k]*Binomial[n, k-1]/n; T[n_, k_]:= Nara[n-k+2, k+1] - Nara[n-k+1, k]; Table[T[n, k], {n,0,15}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Dec 28 2019 *)

PARI

T(n,k) = my(b=binomial); b(n-k+2,k+1)*b(n-k+2,k)/(n-k+2) - b(n-k+1,k)* b(n-k+1,k-1)/(n-k+1);
for(n=0,15, for(k=0, n\2, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 28 2019

Sage

b=binomial;
def T(n,k): return b(n-k+2,k+1)*b(n-k+2,k)/(n-k+2) - b(n-k+1,k)* b(n-k+1,k-1)/(n-k+1)
[[T(n,k) for k in (0..floor(n/2))] for n in (0..15)] # G. C. Greubel, Dec 28 2019

Formula

T(n, k) = Narayana(n-k+2, k+1) - Narayana(n-k+1, k) where Narayana(n, k) = binomial(n, k)*binomial(n, k-1)/n is A001263.

G.f.: G=G(t,z) satisfies t*z^4*G^2 - (1 - z - 2*t*z^2 - t*z^3 + t^2*z^4)*G + 1 = 0. - Emeric Deutsch, Sep 18 2014