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%I M4795 N2047 #114 Jun 11 2023 11:25:51
%S 1,11,66,302,1191,4293,14608,47840,152637,478271,1479726,4537314,
%T 13824739,41932745,126781020,382439924,1151775897,3464764515,
%U 10414216090,31284590870,93941852511,282010106381,846416194536,2540053889352,7621839388981,22869007827143
%N Eulerian numbers (Euler's triangle: column k=3 of A008292, column k=2 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] with exactly 2 descents. - _Mike Zabrocki_, Nov 10 2004
%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. 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.
%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 Vincenzo Librandi, <a href="/A000460/b000460.txt">Table of n, a(n) for n = 3..1000</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 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 Wayne A. Johnson, <a href="https://arxiv.org/abs/2303.16991">An Euler operator approach to Ehrhart series</a>, arXiv:2303.16991 [math.CO], 2023.
%H J. C. P. Miller, <a href="/A002439/a002439_1.pdf">Letter to N. J. A. Sloane, Mar 26 1971</a>
%H O. J. Munch, <a href="/A000460/a000460.pdf">Om potensproduktsummer</a> [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]
%H O. J. Munch, <a href="http://www.jstor.org/stable/24524919">Om potensproduktsummer</a> [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.
%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, pp. 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 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H J. Riordan, <a href="/A000217/a000217_2.pdf">Review of Frankel (1950)</a> [Annotated scanned copy]
%H Sittipong Thamrongpairoj, <a href="https://escholarship.org/uc/item/7j561211">Dowling Set Partitions, and Positional Marked Patterns</a>, Ph. D. Dissertation, University of California-San Diego (2019).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>
%H Robert G. Wilson v, <a href="/A007347/a007347.pdf">Letter to N. J. A. Sloane, Apr. 1994</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10, -40, 82, -91, 52, -12).
%F a(n+2) = 3^(n+2) - (n+3)*2^(n+2) + (1/2)*(n+2)*(n+3). - _Randall L Rathbun_, Jan 22 2002
%F G.f.: x^3*(1+x-4*x^2)/((1-x)^3*(1-2*x)^2*(1-3*x)). - _Mike Zabrocki_, Nov 10 2004
%F a(n) = 3^n - (n+1)*2^n + (1/2)*n*(n+1). - _Gary Detlefs_, Nov 11 2011
%F E.g.f.: exp(x)*(exp(2*x) - (1 + 2*x)*exp(x) + x + x^2/2). - _Wolfdieter Lang_, Apr 17 2017
%p A000460:=-z*(-1-z+4*z**2)/(-1+3*z)/(2*z-1)**2/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation
%t 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, 1, k - 1}], {n, 1, 23}] (* or *)
%t Array[3^(# + 2) - (# + 3)*2^(# + 2) + (1/2)*(# + 2)*(# + 3) &, 23] (* _Michael De Vlieger_, Aug 04 2015, after PARI *)
%o (PARI) A000460(n) = 3^(n+2)-(n+3)*2^(n+2)+(1/2)*(n+2)*(n+3)
%o (Magma) [3^n-(n+1)*2^n+(1/2)*n*(n+1): n in [3..30]]; // _Vincenzo Librandi_, Apr 18 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. A000295.
%K nonn,easy
%O 3,2
%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
%E More terms from _Christian G. Bower_, May 12 2000
%E More terms from _Mike Zabrocki_, Nov 10 2004
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