cp's OEIS Frontend

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

%I A000498 M5188 N2255 #98 Feb 16 2025 08:32:21
%S A000498 1,26,302,2416,15619,88234,455192,2203488,10187685,45533450,198410786,
%T A000498 848090912,3572085255,14875399450,61403313100,251732291184,
%U A000498 1026509354985,4168403181210,16871482830550,68111623139600,274419271461131,1103881308184906,4434992805213952
%N A000498 Eulerian numbers (Euler's triangle: column k=4 of A008292, column k=3 of A173018).
%C A000498 There are 2 versions of Euler's triangle:
%C A000498 * A008292 Classic version of Euler's triangle used by Comtet (1974).
%C A000498 * A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).
%C A000498 Euler's triangle rows and columns indexing conventions:
%C A000498 * 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 A000498 * A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0.(Graham et al.)
%C A000498 Number of permutations of n letters with exactly 3 descents.
%D A000498 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 A000498 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
%D A000498 F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
%D A000498 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
%D A000498 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
%D A000498 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000498 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000498 T. D. Noe, <a href="/A000498/b000498.txt">Table of n, a(n) for n = 4..200</a>
%H A000498 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 A000498 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 A000498 F. N. Castro, O. E. González, and 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 A000498 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 A000498 J. C. P. Miller, <a href="/A002439/a002439_1.pdf">Letter to N. J. A. Sloane, Mar 26 1971</a>
%H A000498 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 A000498 P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.
%H A000498 P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
%H A000498 J. Riordan, <a href="/A000217/a000217_2.pdf">Review of Frankel (1950)</a> [Annotated scanned copy]
%H A000498 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>
%H A000498 Robert G. Wilson v, <a href="/A007347/a007347.pdf">Letter to N. J. A. Sloane, Apr. 1994</a>
%H A000498 <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 A000498 From _Mike Zabrocki_, Nov 12 2004: (Start)
%F A000498 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 A000498 a(n) = (6*4^n - 6*(n + 1)*3^n + 3*(n)*(n + 1)*2^n - (n - 1)*(n)*(n + 1))/6. (End)
%F A000498 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 A000498 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 A000498 There is one permutation of 4 with exactly 3 descents (4321).
%e A000498 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 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 A000498 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 A000498 Table[Sum[(-1)^k*Binomial[n+1,k]*(4-k)^n, {k,0,3}], {n,4,50}] (* _G. C. Greubel_, Oct 23 2017 *)
%o A000498 (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 A000498 (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
%o A000498 (Magma) [EulerianNumber(n,3): n in [4..50]]; // _G. C. Greubel_, Dec 07 2024
%o A000498 (SageMath)
%o A000498 from sage.combinat.combinat import eulerian_number
%o A000498 print([eulerian_number(n,3) for n in range(4,61)]) # _G. C. Greubel_, Dec 07 2024
%Y A000498 Cf. A008292 (classic version of Euler's triangle used by Comtet (1974).)
%Y A000498 Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).)
%Y A000498 Cf. A066912.
%K A000498 nonn,nice,easy
%O A000498 4,2
%A A000498 _N. J. A. Sloane_, _Mira Bernstein_, and _Robert G. Wilson v_
%E A000498 More terms from _Christian G. Bower_, May 12 2000