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A000460
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Eulerian numbers (Euler's triangle: column k=3 of A008292, column k=2 of A173018).
(Formerly M4795 N2047)
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17
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1, 11, 66, 302, 1191, 4293, 14608, 47840, 152637, 478271, 1479726, 4537314, 13824739, 41932745, 126781020, 382439924, 1151775897, 3464764515, 10414216090, 31284590870, 93941852511, 282010106381, 846416194536, 2540053889352, 7621839388981, 22869007827143
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OFFSET
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3,2
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COMMENTS
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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] with exactly 2 descents. - Mike Zabrocki, Nov 10 2004
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REFERENCES
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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. 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.
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).
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LINKS
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O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]
O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
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FORMULA
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a(n+2) = 3^(n+2) - (n+3)*2^(n+2) + (1/2)*(n+2)*(n+3). - Randall L Rathbun, Jan 22 2002
G.f.: x^3*(1+x-4*x^2)/((1-x)^3*(1-2*x)^2*(1-3*x)). - Mike Zabrocki, Nov 10 2004
a(n) = 3^n - (n+1)*2^n + (1/2)*n*(n+1). - Gary Detlefs, Nov 11 2011
E.g.f.: exp(x)*(exp(2*x) - (1 + 2*x)*exp(x) + x + x^2/2). - Wolfdieter Lang, Apr 17 2017
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MAPLE
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MATHEMATICA
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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 *)
Array[3^(# + 2) - (# + 3)*2^(# + 2) + (1/2)*(# + 2)*(# + 3) &, 23] (* Michael De Vlieger, Aug 04 2015, after PARI *)
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PROG
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(PARI) A000460(n) = 3^(n+2)-(n+3)*2^(n+2)+(1/2)*(n+2)*(n+3)
(Magma) [3^n-(n+1)*2^n+(1/2)*n*(n+1): n in [3..30]]; // Vincenzo Librandi, Apr 18 2017
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CROSSREFS
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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)).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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