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A001244 Eulerian numbers (Euler's triangle: column k=8 of A008292, column k=7 of A173018)
(Formerly M5457 N2366)
1, 502, 47840, 2203488, 66318474, 1505621508, 27971176092, 447538817472, 6382798925475, 83137223185370, 1006709967915228, 11485644635009424, 124748182104463860, 1300365805079109480, 13093713503185076040 (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.)

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


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. 2601.

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).


Table of n, a(n) for n=8..22.

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


a(n) = 8^(n+8-1)+sum(i=1, 8-1, (-1)^i/i!*(8-i)^(n+8-1)*prod(j=1, i, n+8+1-j)). - Randall L. Rathbun (randallr(AT)abac.com), Jan 23 2002


k = 8; 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) A001244(n)=8^(n+8-1)+sum(i=1, 8-1, (-1)^i/i!*(8-i)^(n+8-1)*prod(j=1, i, n+8+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. A123125 (row reversed version of A173018.

Cf. A000012, A000460, A000498, A000505, A000514, A001243 (columns for smaller k).

Sequence in context: A121577 A250609 A251266 * A160508 A067949 A200959

Adjacent sequences:  A001241 A001242 A001243 * A001245 A001246 A001247




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 August 18 14:29 EDT 2017. Contains 290720 sequences.