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 A047969 #66 Feb 16 2025 08:32:39
%S A047969 1,1,1,1,3,1,1,5,7,1,1,7,19,15,1,1,9,37,65,31,1,1,11,61,175,211,63,1,
%T A047969 1,13,91,369,781,665,127,1,1,15,127,671,2101,3367,2059,255,1,1,17,169,
%U A047969 1105,4651,11529,14197,6305,511,1,1,19,217,1695,9031
%N A047969 Square array of nexus numbers a(n,k) = (n+1)^(k+1) - n^(k+1) (n >= 0, k >= 0) read by upwards antidiagonals.
%C A047969 If each row started with an initial 0 (i.e., a(n,k) = (n+1)^k - n^k) then each row would be the binomial transform of the preceding row. - _Henry Bottomley_, May 31 2001
%C A047969 a(n-1, k-1) is the number of ordered k-tuples of positive integers such that the largest of these integers is n. - _Alford Arnold_, Sep 07 2005
%C A047969 From _Alford Arnold_, Jul 21 2006: (Start)
%C A047969 The sequences in A047969 can also be calculated using the Eulerian Array (A008292) and Pascal's Triangle (A007318) as illustrated below: (cf. A101095).
%C A047969 1 1 1 1 1 1
%C A047969 1 1 1 1 1 1
%C A047969 -----------------------------------------
%C A047969 1 2 3 4 5 6
%C A047969 1 2 3 4 5
%C A047969 1 3 5 7 9 11
%C A047969 -----------------------------------------
%C A047969 1 3 6 10 15 21
%C A047969 4 12 24 40 60
%C A047969 1 3 6 10
%C A047969 1 7 19 37 61 91
%C A047969 -----------------------------------------
%C A047969 1 4 10 20 35 56
%C A047969 11 44 110 220 385
%C A047969 11 44 110 220
%C A047969 1 4 10
%C A047969 1 15 65 175 369 671
%C A047969 ----------------------------------------- (End)
%C A047969 From _Peter Bala_, Oct 26 2008: (Start)
%C A047969 The above remarks of _Alford Arnold_ may be summarized by saying that (the transpose of) this array is the Hilbert transform of the triangle of Eulerian numbers A008292 (see A145905 for the definition of the Hilbert transform). In this context, A008292 is best viewed as the array of h-vectors of permutohedra of type A. See A108553 for the Hilbert transform of the array of h-vectors of type D permutohedra. Compare this array with A009998.
%C A047969 The polynomials n^k - (n-1)^k, k = 1,2,3,..., which give the nonzero entries in the columns of this array, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re s = 1/2 in the complex plane. See A019538 for the connection between the polynomials n^k - (n-1)^k and the Stirling polynomials of the simplicial complexes dual to the type A permutohedra.
%C A047969 (End)
%C A047969 Empirical: (n+1)^(k+1) - n^(k+1) is the number of first differences of length k+1 arrays of numbers in 0..n, k > 0. - _R. H. Hardin_, Jun 30 2013
%C A047969 a(n-1, k-1) is the number of bargraphs of width k and height n. Examples: a(1,2) = 7 because we have [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1], and [2,2,2]; a(2,1) = 5 because we have [1,3], [2,3], [3,1], [3,2], and [3,3] (bargraphs are given as compositions). This comment is equivalent to A. Arnold's Sep 2005 comment. - _Emeric Deutsch_, Jan 30 2017
%D A047969 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.
%H A047969 T. D. Noe, <a href="/A047969/b047969.txt">Rows n = 0..100 of triangle, flattened</a>
%H A047969 A. Blecher, C. Brennan, A. Knopfmacher and Helmut Prodinger, <a href="http://dx.doi.org/10.1016/j.dam.2014.08.026">The height and width of bargraphs</a>, Discrete Applied Math. 180, (2015), 36-44.
%H A047969 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NexusNumber.html">Nexus Number</a>
%F A047969 From _Vladimir Kruchinin_: (Start)
%F A047969 O.g.f. of e.g.f of rows of array: ((1-x)*exp(y))/(1-x*exp(y))^2.
%F A047969 T(n,m) = Sum_{k=0..m} k!*(-1)^(m+k)*Stirling2(m,k)*C(n+k-1,n), T(n,0)=1.(End)
%F A047969 From _Wolfdieter Lang_, May 07 2021: (Start)
%F A047969 T(n,m) = a(n-m,m) = (n-m+1)^(m+1) - (n-m)^(m+1), n >= 0, m = 0, 1,..., n.
%F A047969 O.g.f. column k of the array: polylog(-(k+1), x)*(1-x)/x. See the _Peter Bala_ comment above, and the Eulerian triangle A008292 formula by _Vladeta Jovovic_, Sep 02 2002.
%F A047969 E.g.f. of e.g.f. of row of the array: exp(y)*(1 + x*(exp(y) - 1))*exp(x*exp(y)).
%F A047969 O.g.f. of triangle's exponential row polynomials R(n, y) = Sum_{m=0} T(n, m)*(y^m)/m!: G(x, y) = exp(x*y)*(1 - x)/(1 - x*exp(x*y))^2. (End)
%e A047969 Array a begins:
%e A047969 [n\k][0 1 2 3 4 5 6 ...
%e A047969 [0] 1 1 1 1 1 1 1 ...
%e A047969 [1] 1 3 7 15 31 63 ...
%e A047969 [2] 1 5 19 65 211 ...
%e A047969 [3] 1 7 37 175 ...
%e A047969 ...
%e A047969 Triangle T begins:
%e A047969 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A047969 0: 1
%e A047969 1: 1 1
%e A047969 2: 1 3 1
%e A047969 3: 1 5 7 1
%e A047969 4: 1 7 19 15 1
%e A047969 5: 1 9 37 65 31 1
%e A047969 6: 1 11 61 175 211 63 1
%e A047969 7: 1 13 91 369 781 665 127 1
%e A047969 8: 1 15 127 671 2101 3367 2059 255 1
%e A047969 9: 1 17 169 1105 4651 11529 14197 6305 511 1
%e A047969 10: 1 19 217 1695 9031 31031 61741 58975 19171 1023 1
%e A047969 ... - _Wolfdieter Lang_, May 07 2021
%t A047969 Flatten[Table[n = d - e; k = e; (n + 1)^(k + 1) - n^(k + 1), {d, 0, 100}, {e, 0, d}]] (* _T. D. Noe_, Feb 22 2012 *)
%o A047969 (Maxima)
%o A047969 T(n,m):=if m=0 then 1 else sum(k!*(-1)^(m+k)*stirling2(m,k)*binomial(n+k-1,n),k,0,m); /* _Vladimir Kruchinin_, Jan 28 2018 */
%Y A047969 Cf. A047970.
%Y A047969 Cf. A009998, A108553 (Hilbert transform of array of h-vectors of type D permutohedra), A145904, A145905.
%Y A047969 Row n sequences of array a: A000012, A000225(k+1), A001047(k+1), A005061(k+1), A005060(k+1), A005062(k+1), A016169(k+1), A016177(k+1), A016185(k+1), A016189(k+1), A016195(k+1), A016197(k+1).
%Y A047969 Column k sequences of array a: (nexus numbers): A000012, A005408, A003215, A005917(n+1), A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528.
%Y A047969 Cf. A343237 (row reversed triangle).
%K A047969 nonn,tabl,nice,easy
%O A047969 0,5
%A A047969 _N. J. A. Sloane_