A000505 Eulerian numbers (Euler's triangle: column k=5 of A008292, column k=4 of A173018).

1, 57, 1191, 15619, 156190, 1310354, 9738114, 66318474, 423281535, 2571742175, 15041229521, 85383238549, 473353301060, 2575022097600, 13796160184500, 73008517581444, 382493246941965, 1987497491971605, 10258045633638475

Offset: 5

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 letters with exactly 4 descents. - 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

• G. C. Greubel, Table of n, a(n) for n = 5..1000
• 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]
• J. Riordan, Review of Frankel (1950) [Annotated scanned copy]
• Robert G. Wilson v, Letter to N. J. A. Sloane, Apr. 1994
• Index entries for linear recurrences with constant coefficients, signature (35, -560, 5432, -35714, 168542, -589632, 1556776, -3126949, 4777591, -5506936, 4703032, -2881136, 1195632, -300672, 34560).

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 Euler's triangle).

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

Programs

Magma

[5^(n + 4) - (n + 5)*4^(n + 4) + (1/2)*(n + 4)*(n + 5)*3^(n + 4) - (1/6)*(n + 3)*(n + 4)*(n + 5)*2^(n + 4) + (1/24)*(n + 2)*(n + 3)*(n + 4)*(n + 5): n in [5..25]]; // G. C. Greubel, Oct 23 2017

Mathematica

k = 5; 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, 19}] (* Michael De Vlieger, Aug 04 2015, after PARI at A001243 *)
a[n_] := 5^n - 2^(n-1)*n*(n^2-1)/3 - 4^n*(n+1) + 3^n*n*(n+1)/2 + (n-2)* (n-1)*n*(n+1)/24; Table[a[n], {n, 5, 25}] (* Jean-François Alcover, Feb 09 2016 *)

PARI

A(n)=5^(n+4)-(n+5)*4^(n+4)+1/2*(n+4)*(n+5)*3^(n+4)-1/6*(n+3)*(n+4)*(n+5)*2^(n+4)+1/24*(n+2)*(n+3)*(n+4)*(n+5)

Formula

a(n) = 5^(n + 4) - (n + 5)*4^(n + 4) + (1/2)*(n + 4)*(n + 5)*3^(n + 4) - (1/6)*(n + 3)*(n + 4)*(n + 5)*2^(n + 4) + (1/24)*(n + 2)*(n + 3)*(n + 4)*(n + 5). - Randall L Rathbun, Jan 22 2002

E.g.f.: (1/24)*exp(x)*(x^4 + 8*x^3 + 12*x^2) - 4*exp(2*x)*(2*x^3/3 + 2*x^2 + x) + 3*exp(3*x)*(9*x^2/2 + 6*x + 1) - 8*exp(4*x)*(2*x + 1) + 5*exp(5*x). - Wenjin Woan, Oct 21 2007

G.f.: (1 + 22*x - 244*x^2 + 422*x^3 + 2575*x^4 - 12012*x^5 + 17828*x^6 - 5664*x^7 - 9552*x^8 + 6912*x^9)*(x/(1-x))^5 / Product_{j=1..4} (1 - (6-j)*x)^j. See the recurrence given in an Apr 03 2017 comment on A123125. - Wolfdieter Lang, Apr 03 2017

Extensions

More terms from Christian G. Bower, May 12 2000