A000498 Eulerian numbers (Euler's triangle: column k=4 of A008292, column k=3 of A173018).

1, 26, 302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450, 198410786, 848090912, 3572085255, 14875399450, 61403313100, 251732291184, 1026509354985, 4168403181210, 16871482830550, 68111623139600, 274419271461131, 1103881308184906, 4434992805213952

Offset: 4

Author

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

Keywords

• nonn
• nice
• 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.)

Number of permutations of n letters with exactly 3 descents.

Examples

			There is one permutation of 4 with exactly 3 descents (4321).
There are 26 permutations of 5 with 3 descents: 15432, 21543, 25431, 31542, 32154, 32541, 35421, 41532, 42153, 42531, 43152, 43215, 43251, 43521, 45321, 51432, 52143, 52431, 53142, 53214, 53241, 53421, 54132, 54213, 54231, 54312. - Neven Juric, Jan 21 2010.
		

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

• T. D. Noe, Table of n, a(n) for n = 4..200
• E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
• L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
• F. N. Castro, O. E. González, and L. A. Medina, The p-adic valuation of Eulerian numbers: trees and Bernoulli numbers, 2014.
• E. T. Frankel, A calculus of figurate numbers and finite differences, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]
• J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971
• Nagatomo Nakamura, Pseudo-Normal Random Number Generation via the Eulerian Numbers, Josai Mathematical Monographs, vol. 8, p 85-95, 2015.
• P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
• P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
• J. Riordan, Review of Frankel (1950) [Annotated scanned copy]
• Eric Weisstein's World of Mathematics, Eulerian Number
• Robert G. Wilson v, Letter to N. J. A. Sloane, Apr. 1994
• Index entries for linear recurrences with constant coefficients, signature (20,-175,882,-2835,6072,-8777,8458,-5204,1848,-288).

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. A066912.

Programs

Magma

[(6*4^n -6*(n+1)*3^n +3*n*(n+1)*2^n -(n-1)*n*(n+1))/6: n in [4..50]]; // G. C. Greubel, Oct 23 2017

Magma

[EulerianNumber(n,3): n in [4..50]]; // G. C. Greubel, Dec 07 2024

Maple

A000498:=proc(n); 4^n-(n+1)*3^n+1/2*(n)*(n+1)*2^n-1/6*(n-1)*(n)*(n+1); end:

Mathematica

LinearRecurrence[{20, -175, 882, -2835, 6072, -8777, 8458, -5204, 1848, -288}, {1, 26, 302, 2416, 15619, 88234, 455192, 2203488, 10187685, 45533450}, 30] (* Jean-François Alcover, Feb 09 2016 *)
Table[Sum[(-1)^k*Binomial[n+1,k]*(4-k)^n, {k,0,3}], {n,4,50}] (* G. C. Greubel, Oct 23 2017 *)

PARI

for(n=4,50, print1((6*4^n -6*(n+1)*3^n +3*n*(n+1)*2^n -(n-1)*n*(n+ 1))/6, ", ")) \\ G. C. Greubel, Oct 23 2017

SageMath

from sage.combinat.combinat import eulerian_number
print([eulerian_number(n,3) for n in range(4,61)]) # G. C. Greubel, Dec 07 2024

Formula

From Mike Zabrocki, Nov 12 2004: (Start)

G.f.: x^4*(1 + 6*x - 43*x^2 + 44*x^3 + 52*x^4 - 72*x^5)/((1-x)^4 * (1-2*x)^3 * (1-3*x)^2 * (1-4*x)).

a(n) = (6*4^n - 6*(n + 1)*3^n + 3*(n)*(n + 1)*2^n - (n - 1)*(n)*(n + 1))/6. (End)

If n>3 is prime, then a(n) == 1 (mod n). A generalization: if a_t(n) denote the number of permutations of n letters with exactly t descents (column t+1 of Euler's triangle A008292), then, for prime n>t, we have a(n) == 1 (mod n). - Vladimir Shevelev, Sep 26 2010

E.g.f.: exp(x)*(exp(3*x) - (1 + 3*x)*exp(2*x) + 2*(x + 2*x^2/2!)*exp(x) - x^2/2! - x^3/3!). - Wolfdieter Lang, Apr 17 2017

Extensions

More terms from Christian G. Bower, May 12 2000