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A000498
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Eulerian numbers (Euler's triangle: column k=4 of A008292, column k=3 of A173018)
(Formerly M5188 N2255)
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3
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1, 26, 302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450, 198410786, 848090912, 3572085255, 14875399450, 61403313100, 251732291184, 1026509354985, 4168403181210, 16871482830550, 68111623139600
(list; graph; refs; listen; history; internal format)
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OFFSET
| 4,2
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COMMENTS
| There are 2 versions of Euler's triangle:
* A008292 Classic version of Euler's triangle used by Comtet (1974).
* A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).
Euler's triangle rows and columns indexing conventions:
* A008292 The rows and columns of the Eulerian triangle are both indexed starting from 1. (Classic version: used in the classic books by Riordan and Comtet.)
* A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0.(Graham et al.)
Number of permutations of n letters with exactly 3 descents.
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REFERENCES
| L. Carlitz et al., Permutations and sequences with repetions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
L. Comtet, "Permutations by Number of Rises; Eulerian Numbers." ยง6.5 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 51 and 240-246, 1974.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=4..200
Eric Weisstein's World of Mathematics, Eulerian Number
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FORMULA
| G.f.: x^4*(1+6*x-43*x^2+44*x^3+52*x^4-72*x^5)/((1-x)^4*(1-2*x)^3*(1-3*x)^2*(1-4*x)); a(n) = 4^n-(n+1)*3^n+1/2*(n)*(n+1)*2^n-1/6*(n-1)*(n)*(n+1). - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 12 2004
If n>3 is prime, then a(n)==1(mod n). A generalization. If a_t(n) denote the number of permutations of n letters with exactly t descents (column t+1 of Euler's triangle A008292), then, for prime n>t, we have a(n)==1(mod n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Sep 26 2010]
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EXAMPLE
| There is one permutation of 4 with exactly 3 descents (4321).
There are 26 permutations of 5 with 3 descents: 15432, 21543, 25431, 31542, 32154, 32541, 35421, 41532, 42153, 42531, 43152, 43215, 43251, 43521, 45321, 51432, 52143, 52431, 53142, 53214, 53241, 53421, 54132, 54213, 54231, 54312. - Neven Juric, Jan 21 2010.
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MAPLE
| A000498:=proc(n); 4^n-(n+1)*3^n+1/2*(n)*(n+1)*2^n-1/6*(n-1)*(n)*(n+1); end:
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CROSSREFS
| Cf. A008292 (classic version of Euler's triangle used by Comtet (1974).)
Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).)
Cf. A066912.
Sequence in context: A022718 A014472 * A066912 A015800 A030647 A202292
Adjacent sequences: A000495 A000496 A000497 * A000499 A000500 A000501
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net), May 12 2000
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