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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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11
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1, 26, 302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450, 198410786, 848090912, 3572085255, 14875399450, 61403313100, 251732291184, 1026509354985, 4168403181210, 16871482830550, 68111623139600
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OFFSET
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4,2
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COMMENTS
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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
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L. Comtet, "Permutations by Number of Rises; Eulerian Numbers." §6.5 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
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P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (20,-175,882,-2835,6072,-8777,8458,-5204,1848,-288).
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FORMULA
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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) = (6*4^n - 6*(n + 1)*3^n + 3*(n)*(n + 1)*2^n - (n - 1)*(n)*(n + 1))/6. (End)
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). - Vladimir Shevelev, Sep 26 2010
E.g.f.: exp(x)*(exp(3*x) - (1 + 3*x)*exp(2*x) + 2*(x + 2*x^2/2!)*exp(x) - x^2/2! - x^3/3!). - Wolfdieter Lang, Apr 17 2017
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EXAMPLE
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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
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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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MATHEMATICA
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LinearRecurrence[{20, -175, 882, -2835, 6072, -8777, 8458, -5204, 1848, -288}, {1, 26, 302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450}, 30] (* Jean-François Alcover, Feb 09 2016 *)
Table[(6*4^n - 6*(n + 1)*3^n + 3*(n)*(n + 1)*2^n - (n - 1)*(n)*(n + 1))/6, {n, 4, 50}] (* G. C. Greubel, Oct 23 2017 *)
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PROG
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(PARI) for(n=4, 50, print1((6*4^n - 6*(n + 1)*3^n + 3*(n)*(n + 1)*2^n - (n - 1)*(n)*(n + 1))/6, ", ")) \\ G. C. Greubel, Oct 23 2017
(Magma) [(6*4^n - 6*(n + 1)*3^n + 3*(n)*(n + 1)*2^n - (n - 1)*(n)*(n + 1))/6: n in [4..50]]; // G. C. Greubel, Oct 23 2017
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CROSSREFS
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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).)
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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