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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
18
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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
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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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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.
Sequence in context: A080842 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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njas
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