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A000514 Eulerian numbers (Euler's triangle: column k=6 of A008292, column k=5 of A173018)
(Formerly M5379 N2336)
8

%I M5379 N2336

%S 1,120,4293,88234,1310354,15724248,162512286,1505621508,12843262863,

%T 102776998928,782115518299,5717291972382,40457344748072,

%U 278794377854832,1879708669896492,12446388300682056,81180715002105741

%N Eulerian numbers (Euler's triangle: column k=6 of A008292, column k=5 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.)

%D L. Comtet, "Permutations by Number of Rises; Eulerian Numbers." ยง6.5 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 G. C. Greubel, <a href="/A000514/b000514.txt">Table of n, a(n) for n = 6..1000</a>

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

%F a(n) = 6^(n+6-1) + Sum_{i=1, 6-1} (-1)^i/i!*(6-i)^(n+6-1)*prod(j=1, i, n+6+1-j)). - Randall L. Rathbun (randallr(AT)abac.com), Jan 23 2002

%F E.g.f.: (1/120)*(120*exp(6*x) - 120*(1+5*x)*exp(5*x) + 480*x*(1+2*x)*exp(4*x) - 540*x^2*(1+x)*exp(3*x) + 80*x^3*(2+x)*exp(2*x) - x^4*(5+x)*exp(x)). - _Wenjin Woan_, Oct 25 2007 (Corrected by _G. C. Greubel_, Oct 24 2017)

%F For the general formula for the o.g.f. and e.g.f. see A123125. - _Wolfdieter Lang_, Apr 03 2017

%t k = 6; 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, 17}] (* _Michael De Vlieger_, Aug 04 2015, after PARI *)

%o (PARI) A000514(n)=6^(n+6-1)+sum(i=1,6-1,(-1)^i/i!*(6-i)^(n+6-1)*prod(j=1,i,n+6+1-j))

%o (PARI) x='x+O('x^50); Vec(serlaplace((1/120)*(120*exp(6*x) - 120*(1+5*x)*exp(5*x) + 480*x*(1+2*x)*exp(4*x) -540*x^2*(1+x)*exp(3*x) +80*x^3*(2+x)*exp(2*x) - x^4*(5+x)*exp(x)))) \\ _G. C. Greubel_, Oct 24 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. A123125 (row reversed version of A173018).

%Y Cf. A000012, A000460, A000498, A000505 (columns for smaller k).

%K nonn,easy

%O 6,2

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

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

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Last modified June 4 08:18 EDT 2020. Contains 334825 sequences. (Running on oeis4.)