A000514 Eulerian numbers (Euler's triangle: column k=6 of A008292, column k=5 of A173018).

1, 120, 4293, 88234, 1310354, 15724248, 162512286, 1505621508, 12843262863, 102776998928, 782115518299, 5717291972382, 40457344748072, 278794377854832, 1879708669896492, 12446388300682056, 81180715002105741, 522859244868123336, 3332058336247871041

Offset: 6

Author

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

Keywords

• nonn
• easy

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." §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).

Links

• 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.
• Robert G. Wilson v, Letter to N. J. A. Sloane, Apr. 1994
• Index entries for linear recurrences with constant coefficients, signature (56, -1470, 24052, -275135, 2339340, -15343384, 79518296, -330867999, 1116881584, -3077867318, 6944399940, -12825741073, 19327952588, -23608674132, 23125043824, -17872240112, 10637255232, -4697205696, 1447365888, -277447680, 24883200).

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)).

Cf. A123125 (row reversed version of A173018).

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

Programs

Mathematica

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

Formula

a(n) = 6^(n+6-1) + Sum_{i=1..6-1} ((-1)^i/i!)*(6-i)^(n+6-1)*Product_{j=1..i} (n+6+1-j). - Randall L Rathbun, 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

Extensions

More terms from Christian G. Bower, May 12 2000