login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001244 Eulerian numbers (Euler's triangle: column k=8 of A008292, column k=7 of A173018)
(Formerly M5457 N2366)
2
1, 502, 47840, 2203488, 66318474, 1505621508, 27971176092, 447538817472, 6382798925475, 83137223185370, 1006709967915228, 11485644635009424, 124748182104463860, 1300365805079109480, 13093713503185076040 (list; graph; refs; listen; history; text; internal format)
OFFSET

8,2

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.)

REFERENCES

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).

LINKS

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.

FORMULA

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

MATHEMATICA

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 *)

PROG

(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))

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).)

Sequence in context: A121577 A250609 A251266 * A160508 A067949 A200959

Adjacent sequences:  A001241 A001242 A001243 * A001245 A001246 A001247

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

More terms from Christian G. Bower, May 12 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 24 21:49 EDT 2017. Contains 283998 sequences.