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A000460
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Eulerian numbers (Euler's triangle: column k=3 of A008292, column k=2 of A173018)
(Formerly M4795 N2047)
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4
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1, 11, 66, 302, 1191, 4293, 14608, 47840, 152637, 478271, 1479726, 4537314, 13824739, 41932745, 126781020, 382439924, 1151775897, 3464764515, 10414216090, 31284590870, 93941852511, 282010106381, 846416194536
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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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] with exactly 2 descents. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 10 2004
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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
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Eulerian Number
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FORMULA
| a(n+2) = 3^(n+2) - (n+3)*2^(n+2) + (1/2)*(n+2)*(n+3). - Randall L. Rathbun (randallr(AT)abac.com), Jan 22 2002
G.f.: x^3*(1+x-4*x^2)/((1-x)^3*(1-2*x)^2*(1-3*x)). - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 10 2004
a(n) = 3^n - (n+1)*2^n + (1/2)*n*(n+1). [From Gary Detlefs, Nov 11 2011]
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MAPLE
| A000460:=-z*(-1-z+4*z**2)/(-1+3*z)/(2*z-1)**2/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
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PROG
| (PARI) A000460(n) = 3^(n+2)-(n+3)*2^(n+2)+(1/2)*(n+2)*(n+3)
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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. A000295.
Sequence in context: A008493 A001287 A022576 * A030115 A091929 A058883
Adjacent sequences: A000457 A000458 A000459 * A000461 A000462 A000463
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KEYWORD
| nonn
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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
More terms from Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 10 2004
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