A000460 Eulerian numbers (Euler's triangle: column k=3 of A008292, column k=2 of A173018).

1, 11, 66, 302, 1191, 4293, 14608, 47840, 152637, 478271, 1479726, 4537314, 13824739, 41932745, 126781020, 382439924, 1151775897, 3464764515, 10414216090, 31284590870, 93941852511, 282010106381, 846416194536, 2540053889352, 7621839388981, 22869007827143

Offset: 3

Author

N. J. A. Sloane, Mira Bernstein, 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.)

Number of permutations of [n] with exactly 2 descents. - Mike Zabrocki, Nov 10 2004

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. B. Remmel et al., The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices, Discrete Math., 342 (2019), 2966-2983. See page 2981.
• 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

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• 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.
• E. T. Frankel, A calculus of figurate numbers and finite differences, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]
• Wayne A. Johnson, An Euler operator approach to Ehrhart series, arXiv:2303.16991 [math.CO], 2023.
• J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971
• O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]
• O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.
• Nagatomo Nakamura, Pseudo-Normal Random Number Generation via the Eulerian Numbers, Josai Mathematical Monographs, vol 8, pp. 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]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• J. Riordan, Review of Frankel (1950) [Annotated scanned copy]
• Sittipong Thamrongpairoj, Dowling Set Partitions, and Positional Marked Patterns, Ph. D. Dissertation, University of California-San Diego (2019).
• 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 (10,-40,82,-91,52,-12).

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. A000295.

Programs

Magma

[3^n-(n+1)*2^n+(1/2)*n*(n+1): n in [3..30]]; // Vincenzo Librandi, Apr 18 2017

Magma

[EulerianNumber(n, 2): n in [3..40]]; // G. C. Greubel, Oct 02 2024

Maple

A000460:=-z*(-1-z+4*z**2)/(-1+3*z)/(2*z-1)**2/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Mathematica

k = 3; Table[k^(n+k-1) + Sum[(-1)^i/i!*(k-i)^(n+k-1) * Product[n+k+1-j, {j, 1, i}], {i, k-1}], {n, 23}] (* or *)
Array[3^(# + 2) - (# + 3)*2^(# + 2) + (1/2)*(# + 2)*(# + 3) &, 23] (* Michael De Vlieger, Aug 04 2015, after PARI *)

PARI

A000460(n) = 3^(n+2)-(n+3)*2^(n+2)+(1/2)*(n+2)*(n+3)

SageMath

def A000460(n): return 3^n - (n+1)*2^n + binomial(n+1,2)
[A000460(n) for n in range(3,31)] # G. C. Greubel, Oct 02 2024

Formula

a(n) = 3^n - (n+1)*2^n + (1/2)*n*(n+1). - Randall L Rathbun, Jan 22 2002

G.f.: x^3*(1+x-4*x^2)/((1-x)^3*(1-2*x)^2*(1-3*x)). - Mike Zabrocki, Nov 10 2004

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

Extensions

More terms from Christian G. Bower, May 12 2000

More terms from Mike Zabrocki, Nov 10 2004