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%I M5317 N2310 #72 Jun 11 2023 11:32:08
%S 1,57,1191,15619,156190,1310354,9738114,66318474,423281535,2571742175,
%T 15041229521,85383238549,473353301060,2575022097600,13796160184500,
%U 73008517581444,382493246941965,1987497491971605,10258045633638475
%N Eulerian numbers (Euler's triangle: column k=5 of A008292, column k=4 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 letters with exactly 4 descents. - Neven Juric, Jan 21 2010
%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. 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="/A000505/b000505.txt">Table of n, a(n) for n = 5..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 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 J. Riordan, <a href="/A000217/a000217_2.pdf">Review of Frankel (1950)</a> [Annotated scanned copy]
%H R. G. Wilson, V, <a href="/A007347/a007347.pdf">Letter to N. J. A. Sloane, Apr. 1994</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (35, -560, 5432, -35714, 168542, -589632, 1556776, -3126949, 4777591, -5506936, 4703032, -2881136, 1195632, -300672, 34560).
%F 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
%F 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
%F 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
%t 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 *)
%t 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 *)
%o (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)
%o (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
%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 Euler's triangle).
%Y Cf. A000012, A000460, A000498 (columns for smaller k).
%K nonn,easy
%O 5,2
%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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