A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 3, 0, 0, 1, 5, 6, 1, 0, 0, 1, 6, 10, 4, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Nadia Heninger and N. J. A. Sloane, May 21 2010

Keywords

• nonn
• tabl

Comments

T(n,k) = 0 if k <0 or k > n+1-k.

T(n,k) is the number of binary vectors of length n and weight k containing no pair of adjacent 1's.

Take Pascal's triangle A007318 and push the k-th column downwards by 2k-1 places (k>=1).

Row sums are A000045.

From Emanuele Munarini, May 24 2011: (Start)

Diagonal sums are A000930(n+1).

A sparse subset (or scattered subset) of {1,2,...,n} is a subset never containing two consecutive elements. T(n,k) is the number of sparse subsets of {1,2,...,n} having size k. For instance, for n=4 and k=2 we have the 3 sparse 2-subsets of {1,2,3,4}: 13, 14, 24. (End)

As a triangle, row 2*n-1 consists of the coefficients of Morgan-Voyce polynomial B(n,x), A172431, and row 2*n to the coefficients of Morgan-Voyce polynomial b(n,x), A054142.

Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014

Antidiagonals of the Pascal matrix A007318 read bottom to top, omitting the first antidiagonal. These are also the antidiagonals (omitting the first antidiagonal) read from top to bottom of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Reverse is embedded in A102426. - Tom Copeland, Jul 02 2018

Examples

			Triangle begins:
  [1]
  [1, 1]
  [1, 2, 0]
  [1, 3, 1, 0]
  [1, 4, 3, 0, 0]
  [1, 5, 6, 1, 0, 0]
  [1, 6, 10, 4, 0, 0, 0]
  [1, 7, 15, 10, 1, 0, 0, 0]
  [1, 8, 21, 20, 5, 0, 0, 0, 0]
  [1, 9, 28, 35, 15, 1, 0, 0, 0, 0]
  [1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0]
  [1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0]
  [1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0]
  [1, 13, 66, 165, 210, 126, 28, 1, 0, 0, 0, 0, 0, 0]
  [1, 14, 78, 220, 330, 252, 84, 8, 0, 0, 0, 0, 0, 0, 0]
  [1, 15, 91, 286, 495, 462, 210, 36, 1, 0, 0, 0, 0, 0, 0, 0]
  [1, 16, 105, 364, 715, 792, 462, 120, 9, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  ...
		

Links

• Indranil Ghosh, Rows 0..125, flattened
• Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
• Emanuele Munarini and Norma Zagaglia Salvi, Scattered Subsets, Discrete Mathematics 267 (2003), 213-228.
• Emanuele. Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
• Emanuele Munarini, A combinatorial interpretation of the Chebyshev polynomials, SIAM Journal on Discrete Mathematics, Volume 20, Issue 3 (2006), 649-655.
• Peter J. Olver, The canonical contact form.
• James J. Y. Zhao, Infinite log-concavity and higher order Turán inequality for Speyer's g-polynomial of uniform matroids, arXiv:2409.08085 [math.CO], 2024. See p. 11.

Crossrefs

Cf. A000045, A000930, A007318, A011973 (another version), A218272.

All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011

A172431 and A054142 describe the odd and even lines of the triangle.

Programs

Mathematica

T[n_,k_]:= Binomial[n+1-k,k]; Table[T[n,k],{n,0,12},{k,0,n}]//Flatten (* Stefano Spezia, Sep 16 2024 *)

Maxima

create_list(binomial(n-k+1,k),n,0,20,k,0,n); /* Emanuele Munarini, May 24 2011 */

PARI

T(n,k)=binomial(n+1-k,k) \\ Charles R Greathouse IV, Oct 24 2012