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 A050446 #94 Jun 07 2025 08:17:42
%S A050446 1,1,1,1,2,1,1,3,3,1,1,5,6,4,1,1,8,14,10,5,1,1,13,31,30,15,6,1,1,21,
%T A050446 70,85,55,21,7,1,1,34,157,246,190,91,28,8,1,1,55,353,707,671,371,140,
%U A050446 36,9,1,1,89,793,2037,2353,1547,658,204,45,10,1,1,144,1782,5864,8272,6405,3164,1086,285,55,11,1
%N A050446 Table read by ascending antidiagonals: T(n, m) giving total degree of n-th-order elementary symmetric polynomials in m variables.
%C A050446 T(n, m) is a polynomial of degree n in m. For example, T(2, m) = (m + 1)(m + 2)/2. For the polynomials corresponding to n = 1, 2, ..., 10, see the Cyvin-Gutman reference (p. 143). Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 12 2005
%C A050446 Let LOOP X C_k, k >= 1, be the graph constructed by attaching a loop to each vertex of the cycle graph C_k. Let G_n, n >= 0, be the graph obtained by deleting one edge from LOOP X C_{n+1} while retaining the n + 1 loops; e.g., for n = 4, see the graph G_4 at the top of the page in the Stanley link below. Then T(n, m) equals the number of magic labelings of G_n having magic sum m. (See the second Mathematica program below which requires the "Omega" package authored by Axel Riese and which can be downloaded from the link provided in the article by Andrews et al.) - _L. Edson Jeffery_, Oct 19 2017
%C A050446 For n != 1, T(n, m) is the number of up-down words of length n over an alphabet of size m. - _Sela Fried_, Apr 08 2025
%C A050446 Conjecture: T(n,m) is the number of words of length n over the alphabet [m] such that any pair of adjacent letters sum to at most m + 1. - _John Tyler Rascoe_, Jun 06 2025
%D A050446 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
%D A050446 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 142-144).
%H A050446 G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.uni-linz.ac.at/research/combinat/risc/publications/#ppaule">MacMahon's partition analysis III. The Omega package</a>.
%H A050446 J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
%H A050446 Byeong-Gil Choe and Hyeong-Kwan Ju, <a href="https://arxiv.org/abs/2305.03930">A Recurrence Relation Associated with Unit-Primitive Matrices</a>, arXiv:2305.03930 [math.CO], 2023.
%H A050446 L. Carlitz and R. Scoville, <a href="https://doi.org/10.1215/S0012-7094-72-03964-6">Up-down sequences</a>, Duke Math. J. (39) (1972), 583-598.
%H A050446 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 A050446 Sela Fried, <a href="https://arxiv.org/abs/2503.02005">A formula for the number of up-down words</a>, arXiv:2503.02005 [math.CO], 2025.
%H A050446 Emma L. L. Gao, Sergey Kitaev, and Philip B. Zhang, <a href="https://arxiv.org/abs/1505.04078">Pattern-avoiding alternating words</a>, arXiv:1505.04078 [math.CO], 2015.
%H A050446 Manfred Goebel, <a href="http://dx.doi.org/10.1007/s002000050118">Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials</a>, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
%H A050446 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 A050446 Daeseok Lee and H.-K. 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 A050446 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 A050446 R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973. [Cached copy, with permission] See p. 31.
%H A050446 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 A050446 T(n, m) = T(n, m - 1) + Sum_{k=0..(n-1)/2} T(2*k, m - 1)*T(n - 1 - 2*k, m).
%F A050446 From _Sela Fried_, Apr 08 2025: (Start)
%F A050446 T(n, m) = 1/(2^(n-1)*(2*m+1))*|Sum_{j = 1..m} tan^2(2*j*Pi/(2*m+1))*sec^(n+1)(2*j*Pi/(2*m+1)))|.
%F A050446 G.f. for words of odd length over an alphabet of size m: x*U_{m-1}(1-x^2/2)/V_{m-1}(1-x^2/2),
%F A050446 g.f. for words of even length over an alphabet of size m: 1/V_{m-1}(1-x^2/2),
%F A050446 where U_k(x) and V_k(x) are the Chebyshev polynomials of the second and third kind, respectively. (End)
%e A050446 Array begins:
%e A050446 [0] 1 1 1 1 1 1 1 1 1 1
%e A050446 [1] 1 2 3 4 5 6 7 8 9 10
%e A050446 [2] 1 3 6 10 15 21 28 36 45 55
%e A050446 [3] 1 5 14 30 55 91 140 204 285 385
%e A050446 [4] 1 8 31 85 190 371 658 1086 1695 2530
%e A050446 [5] 1 13 70 246 671 1547 3164 5916 10317 17017
%e A050446 [6] 1 21 157 707 2353 6405 15106 31998 62349 113641
%e A050446 [7] 1 34 353 2037 8272 26585 72302 173502 377739 760804
%e A050446 [8] 1 55 793 5864 29056 110254 345775 940005 2286648 5089282
%e A050446 [9] 1 89 1782 16886 102091 457379 1654092 5094220 13846117 34053437
%e A050446 ...
%e A050446 Triangle starts:
%e A050446 [0] 1;
%e A050446 [1] 1, 1;
%e A050446 [2] 1, 2, 1;
%e A050446 [3] 1, 3, 3, 1;
%e A050446 [4] 1, 5, 6, 4, 1;
%e A050446 [5] 1, 8, 14, 10, 5, 1;
%e A050446 [6] 1, 13, 31, 30, 15, 6, 1;
%p A050446 A050446 := proc(n,m)
%p A050446 option remember;
%p A050446 if m=0 then
%p A050446 1;
%p A050446 else
%p A050446 procname(n,m-1)+add( procname(2*k,m-1) *procname(n-1-2*k,m), k=0..floor((n-1)/2) );
%p A050446 end if;
%p A050446 end proc:
%p A050446 for d from 0 to 12 do
%p A050446 for m from 0 to d do
%p A050446 printf("%d,",A050446(d-m,m)) ;
%p A050446 end do:
%p A050446 end do: # _R. J. Mathar_, Dec 14 2011
%p A050446 A050446 := := (n, m) -> evalf(abs(add(tan(2*j*Pi/(2*m + 1))^2*sec(2*j*Pi/(2*m + 1))^(n - 1), j = 1 .. m))/(2^(n - 1)*(2*m + 1))): # _Sela Fried_, Apr 28 2025
%t A050446 t[n_, m_?Positive] := t[n, m] = t[n, m-1] + Sum[t[2k, m-1]*t[n-1 - 2k, m], {k, 0, (n-1)/2}]; t[n_, 0] = 1; Flatten[Table[t[i-k , k-1], {i, 1, 12}, {k, 1, i}]] (* _Jean-François Alcover_, Jul 25 2011, after formula *)
%t A050446 << Omega.m; nmax = 9; Do[cond[n_] = {}; If[n == 0, cond[n] = {a[1] == a[2]}, AppendTo[cond[n], {a[1] + a[2] == a[2 n + 2], a[2 n] + a[2 n + 1] == a[2 n + 2]}]; If[n > 1, Do[AppendTo[cond[n], a[2 j] + a[2 j + 1] + a[2 j + 2] == a[2 n + 2]], {j, n - 1}]]]; cond[n] = Flatten[cond[n]]; f[n_] = OEqSum[Product[x[i]^a[i], {i, 2 n + 2}], cond[n], u][[1]] /. x[2 n + 2] -> y /. x[_] -> 1; Do[f[n] = OEqR[f[n], Subscript[u, j]], {j, Length[cond[n]]}], {n, 0, nmax}]; Grid[Table[CoefficientList[Series[f[n], {y, 0, nmax}], y], {n, 0, nmax}]] (* _L. Edson Jeffery_, Oct 19 2017 *)
%o A050446 (Python)
%o A050446 from functools import cache
%o A050446 @cache
%o A050446 def T(n, k):
%o A050446 return T(n, k - 1) + sum(T(2 * j, k - 1) * T(n - 1 - 2 * j, k)
%o A050446 for j in range(1 + (n - 1) // 2)) if k > 0 else 1
%o A050446 for n in range(6): print([T(n - k, k) for k in range(n + 1)])
%o A050446 # _Peter Luschny_, Jun 08 2024
%Y A050446 Rows give A000012, A000027, A000217, A000330, A006322, ...
%Y A050446 Columns give A000012, A000045, A000045, A006356, A006357, A006358, ...
%Y A050446 Cf. A050447.
%K A050446 nonn,easy,nice,tabl
%O A050446 0,5
%A A050446 _N. J. A. Sloane_, Dec 23 1999
%E A050446 More terms from _Naohiro Nomoto_, Jul 03 2001