%I A047969
%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,
%T A047969 63,1,1,13,91,369,781,665,127,1,1,15,127,671,2101,3367,2059,255,
%U A047969 1,1,17,169,1105,4651,11529,14197,6305,511,1,1,19,217,1695,9031
%N A047969 Square array of nexus numbers a(n,k)=(n+1)^(k+1)-n^(k+1) (n >= 0, k >
               = 0) read by 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 (se16(AT)btinternet.com), 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 (Alford1940(AT)AOL.com), 
               Sep 07 2005
%C A047969 Comment from Alford Arnold (Alford1940(AT)aol.com), Jul 21 2006: 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 -----------------------------------------
%C A047969 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008: 
               (Start)
%C A047969 The above remarks of Alford Arnold may be summarised 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 non-zero 
               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)
%D A047969 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 
               1996, p. 54.
%H A047969 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NexusNumber.html">Nexus Number</a>
%e A047969 Array begins
%e A047969 1 1 1 1 1 1 1 1 ...
%e A047969 1 3 7 15 31 63 ...
%e A047969 1 5 19 65 211 ...
%e A047969 1 7 37 175 ...
%Y A047969 Cf. A047970. Rows include A005917, A022521, A022522, etc.
%Y A047969 A009998, A108553 (Hilbert transform of array of h-vectors of type D permutohedra), 
               A145904, A145905. [From Peter Bala (pbala(AT)toucansurf.com), Oct 
               26 2008]
%Y A047969 Sequence in context: A145661 A119258 A099608 this_sequence A047812 A129392 
               A118538
%Y A047969 Adjacent sequences: A047966 A047967 A047968 this_sequence A047970 A047971 
               A047972
%K A047969 nonn,tabl,nice,easy
%O A047969 0,5
%A A047969 N. J. A. Sloane (njas(AT)research.att.com).