A107131 A Motzkin related triangle.

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 2, 6, 1, 0, 0, 0, 10, 10, 1, 0, 0, 0, 5, 30, 15, 1, 0, 0, 0, 0, 35, 70, 21, 1, 0, 0, 0, 0, 14, 140, 140, 28, 1, 0, 0, 0, 0, 0, 126, 420, 252, 36, 1, 0, 0, 0, 0, 0, 42, 630, 1050, 420, 45, 1, 0, 0, 0, 0, 0, 0, 462, 2310, 2310, 660, 55, 1

Offset: 0

Author

Paul Barry, May 12 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are Motzkin numbers A001006. Diagonal sums are A025250(n+1).

Inverse binomial transform of Narayana number triangle A001263. - Paul Barry, May 15 2005

T(n,k)=number of Motzkin paths of length n with k steps U=(1,1) or H=(1,0). Example: T(3,2)=3 because we have HUD, UDH and UHD (here D=(1,-1)). T(n,k) = number of bushes with n+1 edges and k+1 leaves (a bush is an ordered tree in which the outdegree of each nonroot node is at least two). - Emeric Deutsch, May 29 2005

Row reverse of A055151. - Peter Bala, May 07 2012

Rows of A088617 are shifted columns of A107131, whose reversed rows are the Motzkin polynomials of A055151, which give the row polynomials (mod signs) of the o.g.f. that is the compositional inverse for an o.g.f. of the Fibonacci polynomials of A011973. The diagonals of A055151 give the rows of A088671, and the antidiagonals (top to bottom) of A088617 give the rows of A107131. The diagonals of A107131 give the columns of A055151. From the relation between A088617 and A107131, the o.g.f. of this entry is (1 - t*x - sqrt((1-t*x)^2 - 4*t*x^2))/(2*t*x^2). - Tom Copeland, Jan 21 2016

Examples

			Triangle begins
  1;
  0,  1;
  0,  1,  1;
  0,  0,  3,  1;
  0,  0,  2,  6,  1;
  0,  0,  0, 10, 10,   1;
  0,  0,  0,  5, 30,  15,   1;
  0,  0,  0,  0, 35,  70,  21,   1;
  0,  0,  0,  0, 14, 140, 140,  28,  1;
  0,  0,  0,  0,  0, 126, 420, 252, 36, 1;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
• Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Consecutive patterns in restricted permutations and involutions, arXiv:1902.02213 [math.CO], 2019.
• Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
• Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.

Crossrefs

Cf. A001006 (row sums), A025250 (diag. sums), A055151 (row reverse).

Cf. A001263, A011973, A025250, A055151, A088617, A088671, A107131.

Programs

Magma

[Binomial(n, 2*(n-k))*Catalan(n-k): k in [0..n], n in [0..13]]; // G. C. Greubel, May 22 2022

Maple

egf := exp(t*x)*hypergeom([],[2],t*x^2);
s := n -> n!*coeff(series(egf,x,n+2),x,n);
seq(print(seq(coeff(s(n),t,j),j=0..n)),n=0..9); # Peter Luschny, Oct 29 2014

Mathematica

T[n_, k_] := Binomial[k+1, n-k+1] Binomial[n, k]/(k+1);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jun 19 2018 *)

SageMath

flatten([[binomial(n, 2*(n-k))*catalan_number(n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 22 2022

Formula

Number triangle T(n, k) = binomial(k+1, n-k+1)*binomial(n, k)/(k+1).

T(n, k) = Sum_{j=0..n} (-1)^(n-j)C(n, j)*C(j+1, k)*C(j+1, k+1)/(j+1). - Paul Barry, May 15 2005

G.f.: G = G(t, z) satisfies G = 1 + t*z*G + t*z^2*G^2. - Emeric Deutsch, May 29 2005

Coefficient array for the polynomials x^n*Hypergeometric2F1((1-n)/2, -n/2; 2; 4/x). - Paul Barry, Oct 04 2008

From Paul Barry, Jan 12 2009: (Start)

G.f.: 1/(1-xy(1+x)/(1-x^2*y/(1-xy(1+x)/(1-x^2y/(1-xy(1+x).... (continued fraction).

T(n,k) = C(n, 2n-2k)*A000108(n-k). (End)