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Search: id:A047969
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| A047969 |
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Square array of nexus numbers a(n,k)=(n+1)^(k+1)-n^(k+1) (n >= 0, k >= 0) read by antidiagonals. |
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+0
22
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| 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, 1, 13, 91, 369, 781, 665, 127, 1, 1, 15, 127, 671, 2101, 3367, 2059, 255, 1, 1, 17, 169, 1105, 4651, 11529, 14197, 6305, 511, 1, 1, 19, 217, 1695, 9031
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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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
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
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).
1.......1.......1.......1.......1.......1
1.......1.......1.......1.......1.......1
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1.......2.......3.......4.......5.......6
........1.......2.......3.......4.......5
1.......3.......5.......7.......9.......11
-----------------------------------------
1.......3.......6.......10.......15.......21
........4.......12.......24......40.......60
.................1.......3........6.......10
1.......7.......19.......37.......61.......91
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1.......4.......10.......20.......35.......56
.......11.......44.......110.......220.......385
................11.......44.......110.......220
..........................1.......4.......10
1.......15.......65.......175.......369.......671
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Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008: (Start)
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.
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.
(End)
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REFERENCES
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J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.
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LINKS
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Eric Weisstein's World of Mathematics, Nexus Number
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EXAMPLE
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Array begins
1 1 1 1 1 1 1 1 ...
1 3 7 15 31 63 ...
1 5 19 65 211 ...
1 7 37 175 ...
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CROSSREFS
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Cf. A047970. Rows include A005917, A022521, A022522, etc.
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]
Sequence in context: A145661 A119258 A099608 this_sequence A047812 A129392 A118538
Adjacent sequences: A047966 A047967 A047968 this_sequence A047970 A047971 A047972
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KEYWORD
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nonn,tabl,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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