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

%I A079487 #59 Apr 04 2025 15:38:29
%S A079487 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,
%T A079487 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,
%U A079487 18,11,5,1,1,6,15,25,35,40,39,32,22,12,5,1,1
%N A079487 Triangle read by rows giving Whitney numbers T(n,k) of Fibonacci lattices.
%C A079487 Row sums are Fibonacci numbers A000045. - _Roger L. Bagula_, Oct 07 2006
%C A079487 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
%H A079487 Giovanni Resta, <a href="/A079487/b079487.txt">Rows n=0..139 of triangle, flattened</a>
%H A079487 Robert G. Donnelly, Molly W. Dunkum, Sasha V. Malone, and Alexandra Nance, <a href="https://arxiv.org/abs/2012.14991">Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras</a>, arXiv:2012.14991 [math.CO], 2020.
%H A079487 A. Khrabrov and K. Kokhas, <a href="http://arxiv.org/abs/1505.06309">Points on a line, shoelace and dominoes</a>, arXiv:1505.06309 [math.CO], (23-May-2015).
%H A079487 Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://arxiv.org/abs/1812.00170">q-deformed rationals and q-continued fractions</a>, arXiv:1812.00170 [math.CO], 2018-2020.
%H A079487 Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://arxiv.org/abs/1908.04365">On q-deformed real numbers</a>, arXiv:1908.04365 [math.QA], 2019.
%H A079487 Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://hal.archives-ouvertes.fr/hal-02270545/">q-deformed rationals and q-continued fractions</a>, (2019) [math].
%H A079487 Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://arxiv.org/abs/2011.10809">Quantum real numbers and q-deformed Conway-Coxeter friezes</a>, arXiv:2011.10809 [math.QA], 2020.
%H A079487 Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://arxiv.org/abs/2503.23834">q-deformed rationals and irrationals</a>, arXiv:2503.23834 [math.CO], 2025. See p. 7.
%H A079487 Emanuele Munarini and Norma Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.
%H A079487 Valentin Ovsienko, <a href="https://amathr.org/wp-content/uploads/2023/03/QOMBINUMReview-1.pdf">Modular invariant q-deformed numbers: first steps</a>, 2023.
%F A079487 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
%e A079487 Triangle begins:
%e A079487 {1},
%e A079487 {1, 1},
%e A079487 {1, 1, 1},
%e A079487 {1, 2, 1, 1},
%e A079487 {1, 2, 2, 2, 1},
%e A079487 {1, 3, 3, 3, 2, 1},
%e A079487 {1, 3, 4, 5, 4, 3, 1},
%e A079487 {1, 4, 6, 7, 7, 5, 3, 1},
%e A079487 {1, 4, 7, 10, 11, 10, 7, 4, 1},
%e A079487 {1, 5, 10, 14, 17, 16, 13, 8, 4, 1},
%e A079487 {1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1}
%t A079487 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 A079487 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 *)
%o A079487 (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 */
%Y A079487 Largest element in each row gives A077419.
%K A079487 nonn,tabl
%O A079487 0,8
%A A079487 _N. J. A. Sloane_, Jan 19 2003
%E A079487 Mma program editing and a(67)-a(79) from _Giovanni Resta_, May 26 2015