This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001243 M5422 N2355 #59 Dec 30 2024 17:03:17
%S A001243 1,247,14608,455192,9738114,162512286,2275172004,27971176092,
%T A001243 311387598411,3207483178157,31055652948388,285997074307300,
%U A001243 2527925001876036,21598596303099900,179385804170146680
%N A001243 Eulerian numbers (Euler's triangle: column k=7 of A008292, column k=6 of A173018).
%C A001243 There are 2 versions of Euler's triangle:
%C A001243 * A008292 Classic version of Euler's triangle used by Comtet (1974).
%C A001243 * A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).
%C A001243 Euler's triangle rows and columns indexing conventions:
%C A001243 * 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 A001243 * A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0. (Graham et al.)
%D A001243 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 A001243 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
%D A001243 F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
%D A001243 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
%D A001243 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
%D A001243 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001243 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001243 Vincenzo Librandi, <a href="/A001243/b001243.txt">Table of n, a(n) for n = 7..1000</a>
%H A001243 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 A001243 R. G. Wilson, V, <a href="/A007347/a007347.pdf">Letter to N. J. A. Sloane, Apr. 1994</a>
%H A001243 <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, signature (84, -3360, 85204, -1538460, 21061260, -227279184, 1984514004, -14280788214, 85828895124, -435042172944, 1872967672764, -6883607484444, 21668771179044, -58531231913904, 135734401224444, -270012108240369, 459750737925864, -667610836187984, 822369705703584, -852988627596768, 737567996531840, -524515347742464, 301116476275200, -135928473663744, 46399971446784, -11247176540160, 1723509964800, -125411328000).
%F A001243 a(n) = 7^(n+7-1) + Sum_{i=1..7-1} ((-1)^i/i!)*(7-i)^(n+7-1) * Product_{j=1..i} (n+7+1 - j). - _Randall L Rathbun_, Jan 23 2002
%F A001243 For the general formula for the o.g.f. and e.g.f. see A123125. - _Wolfdieter Lang_, Apr 19 2017
%F A001243 a(n) = 7^(n+6) + Sum_{j=1..6} (-1)^j*binomial(n+7, j)*(7-j)^(n+6). - _G. C. Greubel_, Dec 30 2024
%t A001243 k = 7; 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, 15}] (* _Michael De Vlieger_, Aug 04 2015, after PARI *)
%o A001243 (PARI) A001243(n)=7^(n+7-1)+sum(i=1,7-1,(-1)^i/i!*(7-i)^(n+7-1)*prod(j=1,i,n+7+1-j))
%o A001243 (Magma) [EulerianNumber(n,6): n in [7..40]]; // _G. C. Greubel_, Dec 30 2024
%o A001243 (SageMath)
%o A001243 # from sage.all import * # (use for Python)
%o A001243 from sage.combinat.combinat import eulerian_number
%o A001243 print([eulerian_number(n,6) for n in range(7,41)]) # _G. C. Greubel_, Dec 30 2024
%Y A001243 Cf. A008292 (classic version of Euler's triangle used by Comtet (1974).)
%Y A001243 Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).)
%Y A001243 Cf. A000012, A000460, A000498, A000505, A000514 (columns for smaller k).
%K A001243 nonn,easy
%O A001243 7,2
%A A001243 _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
%E A001243 More terms from _Christian G. Bower_, May 12 2000