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A000514 Eulerian numbers (Euler's triangle: column k=6 of A008292, column k=5 of A173018)
(Formerly M5379 N2336)
1, 120, 4293, 88234, 1310354, 15724248, 162512286, 1505621508, 12843262863, 102776998928, 782115518299, 5717291972382, 40457344748072, 278794377854832, 1879708669896492, 12446388300682056, 81180715002105741 (list; graph; refs; listen; history; text; internal format)



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.)


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).


G. C. Greubel, Table of n, a(n) for n = 6..1000

L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.

R. G. Wilson, V, Letter to N. J. A. Sloane, Apr. 1994


a(n) = 6^(n+6-1) + Sum_{i=1, 6-1} (-1)^i/i!*(6-i)^(n+6-1)*prod(j=1, i, n+6+1-j)). - Randall L. Rathbun (randallr(AT)abac.com), Jan 23 2002

E.g.f.: (1/120)*(120*exp(6*x) - 120*(1+5*x)*exp(5*x) + 480*x*(1+2*x)*exp(4*x) - 540*x^2*(1+x)*exp(3*x) + 80*x^3*(2+x)*exp(2*x) - x^4*(5+x)*exp(x)). - Wenjin Woan, Oct 25 2007 (Corrected by G. C. Greubel, Oct 24 2017)

For the general formula for the o.g.f. and e.g.f. see A123125. - Wolfdieter Lang, Apr 03 2017


k = 6; Table[k^(n + k - 1) + Sum[(-1)^i/i!*(k - i)^(n + k - 1) * Product[n + k + 1 - j, {j, 1, i}], {i, 1, k - 1}], {n, 1, 17}] (* Michael De Vlieger, Aug 04 2015, after PARI *)


(PARI) A000514(n)=6^(n+6-1)+sum(i=1, 6-1, (-1)^i/i!*(6-i)^(n+6-1)*prod(j=1, i, n+6+1-j))

(PARI) x='x+O('x^50); Vec(serlaplace((1/120)*(120*exp(6*x) - 120*(1+5*x)*exp(5*x) + 480*x*(1+2*x)*exp(4*x) -540*x^2*(1+x)*exp(3*x) +80*x^3*(2+x)*exp(2*x) - x^4*(5+x)*exp(x)))) \\ G. C. Greubel, Oct 24 2017


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. A123125 (row reversed version of A173018).

Cf. A000012, A000460, A000498, A000505 (columns for smaller k).

Sequence in context: A222003 A139389 A166596 * A179060 A055360 A001807

Adjacent sequences:  A000511 A000512 A000513 * A000515 A000516 A000517




N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v


More terms from Christian G. Bower, May 12 2000



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Last modified January 18 01:14 EST 2019. Contains 319258 sequences. (Running on oeis4.)