login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000498 Eulerian numbers (Euler's triangle: column k=4 of A008292, column k=3 of A173018)
(Formerly M5188 N2255)
11

%I M5188 N2255 #84 Sep 08 2022 08:44:28

%S 1,26,302,2416,15619,88234,455192,2203488,10187685,45533450,198410786,

%T 848090912,3572085255,14875399450,61403313100,251732291184,

%U 1026509354985,4168403181210,16871482830550,68111623139600

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

%C There are 2 versions of Euler's triangle:

%C * A008292 Classic version of Euler's triangle used by Comtet (1974).

%C * A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).

%C Euler's triangle rows and columns indexing conventions:

%C * 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.)

%C * A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0.(Graham et al.)

%C Number of permutations of n letters with exactly 3 descents.

%D 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.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.

%D F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.

%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000498/b000498.txt">Table of n, a(n) for n = 4..200</a>

%H E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce <a href="http://arxiv.org/abs/1508.03673">A generalization of Eulerian numbers via rook placements</a>, arXiv:1508.03673 [math.CO], 2015.

%H L. Carlitz et al., <a href="http://dx.doi.org/10.1016/S0021-9800(66)80057-1">Permutations and sequences with repetitions by number of increases</a>, J. Combin. Theory, 1 (1966), 350-374.

%H F. N. Castro, O. E. González, L. A. Medina, <a href="http://emmy.uprrp.edu/lmedina/papers/eulerian/EulerianFinal.pdf">The p-adic valuation of Eulerian numbers: trees and Bernoulli numbers</a>, 2014.

%H E. T. Frankel, <a href="/A000217/a000217_1.pdf"> A calculus of figurate numbers and finite differences</a>, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]

%H J. C. P. Miller, <a href="/A002439/a002439_1.pdf">Letter to N. J. A. Sloane, Mar 26 1971</a>

%H Nagatomo Nakamura, <a href="http://libir.josai.ac.jp/il/user_contents/02/G0000284repository/pdf/JOS-13447777-0808.pdf">Pseudo-Normal Random Number Generation via the Eulerian Numbers</a>, Josai Mathematical Monographs, vol. 8, p 85-95, 2015.

%H P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.

%H P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]

%H J. Riordan, <a href="/A000217/a000217_2.pdf">Review of Frankel (1950)</a> [Annotated scanned copy]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>

%H R. G. Wilson, V, <a href="/A007347/a007347.pdf">Letter to N. J. A. Sloane, Apr. 1994</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (20, -175, 882, -2835, 6072, -8777, 8458, -5204, 1848, -288).

%F From _Mike Zabrocki_, Nov 12 2004: (Start)

%F 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)).

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

%F 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

%F 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

%e There is one permutation of 4 with exactly 3 descents (4321).

%e 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.

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

%t 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 *)

%t Table[(6*4^n - 6*(n + 1)*3^n + 3*(n)*(n + 1)*2^n - (n - 1)*(n)*(n + 1))/6, {n, 4, 50}] (* _G. C. Greubel_, Oct 23 2017 *)

%o (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

%o (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

%Y Cf. A008292 (classic version of Euler's triangle used by Comtet (1974).)

%Y Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).)

%Y Cf. A066912.

%K nonn,nice,easy

%O 4,2

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_

%E More terms from _Christian G. Bower_, May 12 2000

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)