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 A231147 #21 Sep 11 2024 10:07:43
%S A231147 1,1,1,1,1,1,3,1,1,1,1,4,3,4,1,1,1,1,5,4,9,4,5,1,1,1,1,6,5,14,9,14,5,
%T A231147 6,1,1,1,1,7,6,20,14,29,14,20,6,7,1,1,1,1,8,7,27,20,49,29,49,20,27,7,
%U A231147 8,1,1,1,1,9,8,35,27,76,49,99,49,76,27,35,8,9
%N A231147 Array of coefficients of numerator polynomials of the rational function p(n, x + 1/x), where p(n,x) = (x^n - 1)/(x - 1).
%C A231147 From _Gus Wiseman_, Mar 19 2023: (Start)
%C A231147 Also appears to be the number of nonempty subsets of {1,...,n} with median k, where k ranges from 1 to n in steps of 1/2, and the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). For example, row n = 5 counts the following subsets:
%C A231147 {1} {1,2} {2} {1,4} {3} {2,5} {4} {4,5} {5}
%C A231147 {1,3} {2,3} {1,5} {3,4} {3,5}
%C A231147 {1,2,3} {1,2,3,4} {2,4} {1,3,4,5} {1,4,5}
%C A231147 {1,2,4} {1,2,3,5} {1,3,4} {2,3,4,5} {2,4,5}
%C A231147 {1,2,5} {1,3,5} {3,4,5}
%C A231147 {2,3,4}
%C A231147 {2,3,5}
%C A231147 {1,2,4,5}
%C A231147 {1,2,3,4,5}
%C A231147 Central diagonals T(n,(n+1)/2) appear to be A100066 (bisection A006134).
%C A231147 For mean instead of median we have A327481.
%C A231147 For partitions instead of subsets we have A359893, full steps A359901.
%C A231147 Central diagonals T(n,n/2) are A361801 (bisection A079309).
%C A231147 (End)
%H A231147 John Tyler Rascoe, <a href="/A231147/b231147.txt">Rows n = 1..100, flattened</a>
%e A231147 Triangle begins:
%e A231147 1
%e A231147 1 1 1
%e A231147 1 1 3 1 1
%e A231147 1 1 4 3 4 1 1
%e A231147 1 1 5 4 9 4 5 1 1
%e A231147 1 1 6 5 14 9 14 5 6 1 1
%e A231147 1 1 7 6 20 14 29 14 20 6 7 1 1
%e A231147 1 1 8 7 27 20 49 29 49 20 27 7 8 1 1
%e A231147 1 1 9 8 35 27 76 49 99 49 76 27 35 8 9 1 1
%e A231147 First 3 polynomials: 1, 1 + x + x^2, 1 + x + 3*x^2 + x^3 + x^4
%t A231147 z = 60; p[n_, x_] := p[x] = (x^n - 1)/(x - 1); Table[p[n, x], {n, 1, z/4}]; f1[n_, x_] := f1[n, x] = Numerator[Factor[p[n, x] /. x -> x + 1/x]]; Table[Expand[f1[n, x]], {n, 0, z/4}]
%t A231147 Flatten[Table[CoefficientList[f1[n, x], x], {n, 1, z/4}]]
%o A231147 (PARI)
%o A231147 A231147_row(n) = {Vecrev(Vec(numerator((-1+(x+(1/x))^n)/(x+(1/x)-1))))} \\ _John Tyler Rascoe_, Sep 10 2024
%Y A231147 Cf. A231148.
%Y A231147 Row sums are 2^n-1 = A000225(n).
%Y A231147 Row lengths are 2n-1 = A005408(n-1).
%Y A231147 Removing every other column appears to give A013580.
%Y A231147 Cf. A006134, A007318, A008289, A079309, A325347, A327481, A359907, A360005.
%K A231147 nonn,tabf,easy
%O A231147 1,7
%A A231147 _Clark Kimberling_, Nov 05 2013