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A001243 Eulerian numbers (Euler's triangle: column k=7 of A008292, column k=6 of A173018)
(Formerly M5422 N2355)
1, 247, 14608, 455192, 9738114, 162512286, 2275172004, 27971176092, 311387598411, 3207483178157, 31055652948388, 285997074307300, 2527925001876036, 21598596303099900, 179385804170146680 (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).


Vincenzo Librandi, Table of n, a(n) for n = 7..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) = 7^(n+7-1)+sum(i=1, 7-1, (-1)^i/i!*(7-i)^(n+7-1)*prod(j=1, i, n+7+1-j)). - Randall L. Rathbun (randallr(AT)abac.com), Jan 23 2002

For the general formula for the o.g.f. and e.g.f. see A123125. - Wolfdieter Lang, Apr 19 2017


k = 7; 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, 15}] (* Michael De Vlieger, Aug 04 2015, after PARI *)


(PARI) A001243(n)=7^(n+7-1)+sum(i=1, 7-1, (-1)^i/i!*(7-i)^(n+7-1)*prod(j=1, i, n+7+1-j))


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. A000012, A000460, A000498, A000505, A000514 (columns for smaller k).

Sequence in context: A166399 A129133 A251265 * A048901 A223546 A187398

Adjacent sequences:  A001240 A001241 A001242 * A001244 A001245 A001246




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 March 19 12:39 EDT 2018. Contains 300853 sequences. (Running on oeis4.)