A079487 Triangle read by rows giving Whitney numbers T(n,k) of Fibonacci lattices.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 3, 2, 1, 1, 3, 4, 5, 4, 3, 1, 1, 4, 6, 7, 7, 5, 3, 1, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 5, 10, 14, 17, 16, 13, 8, 4, 1, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 1, 6, 15, 25, 35, 40, 39, 32, 22, 12, 5, 1, 1

Offset

0,8

Comments

Row sums are Fibonacci numbers A000045. - Roger L. Bagula, Oct 07 2006

This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008

Links

Giovanni Resta, Rows n=0..139 of triangle, flattened

Robert G. Donnelly, Molly W. Dunkum, Sasha V. Malone, and Alexandra Nance, Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras, arXiv:2012.14991 [math.CO], 2020.

A. Khrabrov and K. Kokhas, Points on a line, shoelace and dominoes, arXiv:1505.06309 [math.CO], (23-May-2015).

Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, arXiv:1812.00170 [math.CO], 2018-2020.

Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365 [math.QA], 2019.

Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, (2019) [math].

Sophie Morier-Genoud and Valentin Ovsienko, Quantum real numbers and q-deformed Conway-Coxeter friezes, arXiv:2011.10809 [math.QA], 2020.

E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Valentin Ovsienko, Modular invariant q-deformed numbers: first steps, 2023.

Formula

Define polynomials by: if k is odd then p(k, x) = x*p(k - 1, x) + p(k - 2, x); if k is even then: p(k, x) = p(k - 1, x) + x^2*p(k - 2, x). Triangle gives array of coefficients. - Roger L. Bagula, Oct 07 2006

Example

Triangle begins:
{1},
{1, 1},
{1, 1, 1},
{1, 2, 1, 1},
{1, 2, 2, 2, 1},
{1, 3, 3, 3, 2, 1},
{1, 3, 4, 5, 4, 3, 1},
{1, 4, 6, 7, 7, 5, 3, 1},
{1, 4, 7, 10, 11, 10, 7, 4, 1},
{1, 5, 10, 14, 17, 16, 13, 8, 4, 1},
{1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1}

Mathematica

p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = Expand@ If[Mod[k, 2] == 1, x*p[k - 1, x] + p[k - 2, x], p[k - 1, x] + x^2*p[k - 2, x]]; Flatten[ Table[CoefficientList[p[n, x], x], {n, 0, 10}]] (* Roger L. Bagula, Oct 07 2006 *)
T[ n_, k_] := (T[n, k] = Which[k<0 || k>n, 0, k==0, 1, True, T[n-1, k-Mod[n, 2]] + T[n-2, k-Mod[n+1, 2]*2]]); (* Michael Somos, Dec 12 2023 *)

Prog

(PARI) {T(n, k) = if(k<0 || k>n, 0, k==0, 1, T(n-1, k-(n%2)) + T(n-2, k-(n+1)%2*2))}; /* Michael Somos, Dec 12 2023 */

Crossrefs

Largest element in each row gives A077419.

Sequence in context: A029339 A029364 A122586 * A229122 A069010 A353332

Adjacent sequences: A079484 A079485 A079486 * A079488 A079489 A079490

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 19 2003

Extensions

Mma program editing and a(67)-a(79) from Giovanni Resta, May 26 2015

Status

approved