This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A205810 #34 Jun 26 2025 07:35:44
%S A205810 1,1,1,1,1,2,1,2,1,1,3,3,4,3,3,1,1,4,6,8,9,8,6,4,1,1,5,10,15,20,21,20,
%T A205810 15,10,5,1,1,6,15,26,39,48,52,48,39,26,15,6,1,1,7,21,42,70,98,119,127,
%U A205810 119,98,70,42,21,7,1
%N A205810 Irregular triangle read by rows: Whitney numbers c_{n,k} (n >= 0, 0 <= k <= 2n) of Lucas lattices.
%H A205810 Mark E. AlSukaiti and Nafaa Chbili, <a href="https://arxiv.org/abs/2303.11398">Alexander and Jones Polynomials of weaving 3-braid links and Whitney rank polynomials of Lucas lattice</a>, arXiv:2303.11398 [math.GT], 2023.
%H A205810 E. Munarini and N. 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.
%F A205810 c(n, k) = n*Sum_{i = 0..floor(k/2)} 1/(n-i)*binomial(n-i, n-k+i)*binomial(k-i-1, i) for 0 <= k <= 2*n-1; c(n, 2*n) = 1. - _Leonid Bedratyuk_, May 15 2018
%F A205810 From _Peter Bala_, Jun 26 2025: (Start)
%F A205810 For n >= 1, the n-th row polynomial R(n, x) = x^n * t(n, 1 + x + 1/x), where t(n, x) = 2*Chebyshev_T(n, x/2) (AlSukaiti and Chbili, Proposition 2.1).
%F A205810 Conjecture: for n >= 1, t(n, x + 1) = Sum_{k = 0..n} c(n, n-k)*t(k, x) - c(n, n). Cf. A097724. (End)
%e A205810 Triangle begins:
%e A205810 1;
%e A205810 1, 1, 1;
%e A205810 1, 2, 1, 2, 1;
%e A205810 1, 3, 3, 4, 3, 3, 1;
%e A205810 1, 4, 6, 8, 9, 8, 6, 4, 1;
%e A205810 1, 5, 10, 15, 20, 21, 20, 15, 10, 5, 1;
%e A205810 1, 6, 15, 26, 39, 48, 52, 48, 39, 26, 15, 6, 1;
%e A205810 1, 7, 21, 42, 70, 98, 119, 127, 119, 98, 70, 42, 21, 7, 1;
%e A205810 ...
%p A205810 c:= (n, k)-> `if`(k=2*n, 1, n*add(1/(n-i)*binomial(n-i, n-k+i)*binomial(k-i-1, i), i=0..floor(k/2))): seq(seq(c(n, k), k=0..2*n), n=0..8); # _Leonid Bedratyuk_, May 15 2018
%o A205810 (PARI) T(n,k) = if (k==2*n, 1, n*sum(i=0, k\2, 1/(n-i)*binomial(n-i,n-k+i)*binomial(k-i-1,i)));
%o A205810 tabf(nn) = for (n=0, nn, for (k=0, 2*n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 16 2018
%Y A205810 Main diagonal is A051292.
%Y A205810 Cf. A079487, A097724.
%K A205810 nonn,tabf
%O A205810 0,6
%A A205810 _N. J. A. Sloane_, Jan 31 2012