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 A205497 #117 Jul 30 2025 17:48:29
%S A205497 1,1,1,1,1,1,3,1,1,7,7,1,1,14,31,14,1,1,26,109,109,26,1,1,46,334,623,
%T A205497 334,46,1,1,79,937,2951,2951,937,79,1,1,133,2475,12331,20641,12331,
%U A205497 2475,133,1,1,221,6267,47191,123216,123216,47191,6267,221,1
%N A205497 Triangle read by rows: Zig-zag Eulerian number triangle T(n, k).
%C A205497 From _Kyle Petersen_, Jun 02 2024: (Start)
%C A205497 Coefficients of the "P-Eulerian" polynomial of a naturally labeled zig-zag poset, which counts linear extensions according to number of descents. T(n, k) is the number of linear extensions of the n-element zig-zag poset with k descents.
%C A205497 Also T(n, k) is the number of up-down permutations of length n with k "big returns". A big return is a pair (i, i+1) for which i appears more than one place to the right of i+1 in the permutation. This interpretation implies row sums are given by A000111. (End)
%C A205497 From _L. Edson Jeffery_, Jan 27 2012: (Start)
%C A205497 (Previous name:) Omitting the first two ones, a rectangular array M read by antidiagonals in which entry M_{n-k, k} in row n-k and column k, 0 <= k <= n, gives the coefficient of x^k in the numerator of the conjectured generating function for row n + 3 of the tabular form of A050446.
%C A205497 In the following, let M_{n, k} denote the entry in row n and column k of M, n, k in {0, 1, ...}.
%C A205497 Conjecture: 1. M_{n, k} = M_{k, n}, for all n and k; that is, M is symmetric about the central terms {1, 3, 31, 623,...}. (This has been verified for the first 100 antidiagonals of M.)
%C A205497 Conjecture: 2. For m in {3, 4,...}, row m of array A050446 has generating function of the form H_m(x)/(1 - x)^m, in which the numerator H_m(x) is a polynomial of degree m - 3 in x with coefficients given by the entries of the (m - 3)-th antidiagonal of M containing the sequence of entries {M_{m-3-j,j}}, j=0..m-3 (see the example below). It is known that H_1(x) = H_2(x) = 1.
%C A205497 Conjecture: 3. Define the Chebyshev polynomials of the second kind by U_0(t) = 1, U_1(t) = 2*t and U_r(t) = 2*t*U_(r-1)(t) - U_(r-2)(t) (r > 1). Assuming Conjecture 1, lim_{n -> infinity} M_{n+1, k}/M_{n, k} = U_k(cos(Pi/(2*k+3))) = spectral radius of the (k+1) X (k+1) unit-primitive matrix (see [Jeffery]) A_{2*k+3, k} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], with identical limits for the columns of the transpose M^T of M.
%C A205497 Conjecture: 4. Let S(u, v) denote the entry in row u and column v of triangle S = A187660, 0 <= v <= u. Define the polynomials P_u(x) = Sum[S(u, v)*x^v]. Assuming Conjecture 1, then (i) the generating function for row (or column) n of M is of the form
%C A205497 G_n(x)/((P_1(x))^(n+1) * (P_2(x))^n * ... * (P_n(x))^2 * P_(n+1)(x)),
%C A205497 in which (ii) the numerator G_n(x) is a polynomial of degree A005586(n), and (iii) the denominator is a polynomial of degree A000292(n+1).
%C A205497 Remarks: The coefficients in the numerators G_n(x) appear to have no pattern. The polynomial P_j(x), j in {1,...,n+1}, of Conjecture 4 is also obtained from the characteristic polynomial of the unit-primitive matrix A_{2*j+3,j} of Conjecture 3 by taking the exponents of the latter in reverse order; and P_j(x) is otherwise identical to the characteristic polynomial of the unit-primitive matrix A_{2*j+3,1}.
%C A205497 (End)
%C A205497 Conjecture: The Eulerian zig-zag polynomials have only negative and simple real roots and form a Sturm sequence, that is, p(n+1, x) has n real roots separated by the roots of p(n, x). This property was proved by Frobenius for the classical Eulerian polynomials. - _Peter Luschny_, Jun 04 2024
%H A205497 Jane Ivy Coons and Seth Sullivant, <a href="https://arxiv.org/abs/1901.07443">The h*-polynomial of the order polytope of the zig-zag poset</a>, arXiv:1901.07443 [math.CO], 2019.
%H A205497 L. Edson Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A205497 Hyeong-Kwan Ju, <a href="https://doi.org/10.5831/HMJ.2017.39.4.665">On the sequence generated by a certain type of matrices</a>, Honam Math. J. 39, No. 4, 665-675 (2017).
%H A205497 Daeseok Lee and Hyeong-Kwan Ju, <a href="http://arxiv.org/abs/1503.05658">An Extension of Hibi's palindromic theorem</a>, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
%H A205497 Peter Luschny, <a href="/A205497/a205497.png">Illustrating the polynomials</a>.
%H A205497 T. Kyle Petersen and Yan Zhuang, <a href="https://arxiv.org/abs/2403.07181">Zig-zag Eulerian polynomials</a>, arXiv:2403.07181 [math.CO], 2024.
%H A205497 Igor Pak, Boris Shapiro, Ilya Smirnov, and Ken-ichi Yoshida, <a href="https://staff.math.su.se/shapiro/Articles/Hilbert-Kunz.pdf">Hilbert-Kunz multiplicity of quadrics via the Ehrhart theory</a>, Stockholm Univ. (Sweden, 2025). See p. 5.
%H A205497 R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973 [Cached copy, with permission] See p. 31.
%H A205497 Guoce Xin and Yueming Zhong, <a href="https://arxiv.org/abs/2201.02376">Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials</a>, arXiv:2201.02376 [math.CO], 2022.
%F A205497 Conjecture: 5.1. G.f. for column 0 of M is 1/(1-x) (A000012).
%F A205497 Conjecture: 5.2. G.f. for column 1 of M is 1/((1-x)^2*(1-x-x^2)) (A001924).
%F A205497 Conjecture: 5.3. G.f. for column 2 of M is (1 - x^2 - x^3 - x^4 + x^5)/((1-x)^3*(1-x-x^2)^2*(1 - 2*x - x^2 + x^3)) (A205492).
%F A205497 Conjecture: 5.4. G.f. for column 3 of M is (1 + x - 6*x^2 - 15*x^3 + 21*x^4 + 35*x^5 - 13*x^6 - 51*x^7 + 3*x^8 + 21*x^9 + 5*x^10 + x^11 - 5*x^12 - x^13 - x^14)/((1-x)^4*(1-x-x^2)^3*(1 - 2*x - x^2 + x^3)^2*(1 - 2*x - 3*x^2 + x^3 + x^4)) (A205493).
%F A205497 Conjecture: 5.5. G.f. for column 4 of M is (1 + 4*x - 31*x^2 - 67*x^3 + 348*x^4 + 418*x^5 - 1893*x^6 - 1084*x^7 + 4326*x^8 + 4295*x^9 - 7680*x^10 - 9172*x^11 + 9104*x^12 + 11627*x^13 - 5483*x^14 - 10773*x^15 + 1108*x^16 + 7255*x^17 + 315*x^18 - 3085*x^19 - 228*x^20 + 669*x^21 + 102*x^22 - 23*x^23 - 45*x^24 - 16*x^25 + 11*x^26 + 2*x^27 - x^28)/((1-x)^5*(1-x-x^2)^4*(1 - 2*x - x^2 + x^3)^3*(1 - 2*x - 3*x^2 + x^3 + x^4)^2*(1 - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5)) (A205494).
%e A205497 From _Kyle Petersen_, Jun 02 2024: (Start)
%e A205497 Triangle T(n, k) begins:
%e A205497 1;
%e A205497 1;
%e A205497 1;
%e A205497 1, 1;
%e A205497 1, 3, 1;
%e A205497 1, 7, 7, 1;
%e A205497 1, 14, 31, 14, 1;
%e A205497 1, 26, 109, 109, 26, 1;
%e A205497 1, 46, 334, 623, 334, 46, 1;
%e A205497 1, 79, 937, 2951, 2951, 937, 79, 1;
%e A205497 ...
%e A205497 For n=4, the naturally labeled zig-zag poset 1<3>2<4 has five linear extensions: 1234, 1243, 2134, 2143, 2413, and their descent numbers are (respectively) 0, 1, 1, 2, 1. Thus T(4,0) = 1, T(4,1) = 3, and T(4,2) = 1. Also with n=4, there are five up-down permutations: 1324, 1423, 2314, 2413, 3412, and their big return numbers are (respectively) 0, 1, 1, 2, 1. (End)
%e A205497 Without the first two ones the data can be seen as an array M read by antidiagonals. Christopher H. Gribble kindly calculated the first 100 antidiagonals which starts as:
%e A205497 1, 1, 1, 1, 1, 1, ...
%e A205497 1, 3, 7, 14, 26, 46, ...
%e A205497 1, 7, 31, 109, 334, 937, ...
%e A205497 1, 14, 109, 623, 2951, 12331, ...
%e A205497 1, 26, 334, 2951, 20641, 123216, ...
%e A205497 1, 46, 937, 12331, 123216, 1019051, ...
%e A205497 ...
%e A205497 The antidiagonals of M written as the rows of a triangle, yielding then, by the conjectures and the definition of H_m(x), row m = 7 of table A050446 has generating function H_7(x)/(1-x)^7 = (Sum_{j=0..4} M_{4-j,j}*x^j)/(1-x)^7 = (1 + 14*x + 31*x^2 + 14*x^3 + x^4)/(1-x)^7.
%p A205497 Gn := proc(n) local F;
%p A205497 if n = 0 then p*q*x/(1 - q*x);
%p A205497 elif n > 0 then
%p A205497 F := Gn(n - 1);
%p A205497 simplify(p/(p - q)*(subs({p = q, q = p}, F) - subs(p = q, F)));
%p A205497 fi;
%p A205497 end:
%p A205497 Zn := proc(n) expand(simplify(subs({p = 1, q = 1}, Gn(n))*(1 - x)^(n + 1))) end:
%p A205497 seq( coeffs(Zn(n)), n=0..15); # _Kyle Petersen_, Jun 02 2024
%p A205497 # Alternative:
%p A205497 A205497row := proc(n) local k, j; ifelse(n < 2, 1,
%p A205497 seq(add((-1)^j * binomial(n + 1, j) * A050446(n, k - j), j = 0..k), k = 0..n-2)) end: # _Peter Luschny_, Jun 17 2024
%t A205497 Gn[n_] := Module[{F}, If[n == 0, p*q*x/(1-q*x), If[n > 0, F = Gn[n-1]; Simplify[p/(p-q)*(ReplaceAll[F, {p -> q, q -> p}] - ReplaceAll[F, p -> q])]]]];
%t A205497 Zn[n_] := Expand[Simplify[ReplaceAll[Gn[n], {p -> 1, q -> 1}]*(1-x)^(n+1)]];
%t A205497 Table[Rest@CoefficientList[Zn[n], x], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Jun 04 2024, after _Kyle Petersen_ *)
%o A205497 (Python)
%o A205497 from functools import cache
%o A205497 from math import comb as binomial
%o A205497 @cache
%o A205497 def S(n, k):
%o A205497 return (S(n, k - 1) + sum(S(2 * j, k - 1) * S(n - 1 - 2 * j, k)
%o A205497 for j in range(1 + (n - 1) // 2)) if k > 0 else 1)
%o A205497 def A205497(dim): # returns [row(0), ..., row(dim-1)]
%o A205497 if dim < 4: return [[1]] * dim
%o A205497 Y = [[0 for _ in range(n - 2)] for n in range(dim + 1)]
%o A205497 for n in range(dim + 1):
%o A205497 for k in range(n - 2):
%o A205497 for j in range(k + 1):
%o A205497 Y[n][k] += (-1)**j * binomial(n, j) * S(n - 1, k - j)
%o A205497 Y[1] = Y[2] = [1]
%o A205497 return Y[1::]
%o A205497 print(A205497(9)) # _Peter Luschny_, Jun 14 2024
%Y A205497 Cf. A000012, A000292, A001924, A050446, A050447, A187660, row sums are A000111.
%Y A205497 Cf. A350354, A373388, A373389.
%K A205497 nonn,tabf
%O A205497 0,7
%A A205497 _L. Edson Jeffery_, Jan 27 2012
%E A205497 Two 1's prepended and new name by _Kyle Petersen_ Jun 02 2024
%E A205497 Edited by _Peter Luschny_, Jun 02 2024