%I M5457 N2366 #64 Jul 30 2024 02:53:28
%S 1,502,47840,2203488,66318474,1505621508,27971176092,447538817472,
%T 6382798925475,83137223185370,1006709967915228,11485644635009424,
%U 124748182104463860,1300365805079109480,13093713503185076040
%N Eulerian numbers (Euler's triangle: column k=8 of A008292, column k=7 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 For the general computation of the o.g.f. and e.g.f. see A123125. - _Wolfdieter Lang_, Apr 03 2017
%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. 2601.
%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="/A001244/b001244.txt">Table of n, a(n) for n = 8..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>
%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, signature (120, -6930, 256564, -6843837, 140161164, -2293167668, 30793317984, -346027498674, 3301174490432, -27034426023228, 191677191769368, -1184495927428914, 6413285791562760, -30547549870770240, 128399094121475760, -477325107218885805, 1571764443755152680, -4588173158058601250, 11875425392771515860, -27240699344951953809, 55318442559624109580, -99273350219483495580, 157041371328829338576, -218253110396224153888, 265336916554318663296, -280638192440433919872, 256449901319079809536, -200704456428999204096, 133025721255740648448, -73584771640934648832, 33313567375875428352, -12012672014150270976, 3315383509586411520, -657169361790566400, 83234996748288000, -5056584744960000).
%F a(n) = 8^(n+8-1) + Sum_{i=1..8-1} ((-1)^i/i!)*(8-i)^(n+8-1) * Product_{j=1..i} (n+8+1 - j). - _Randall L Rathbun_, Jan 23 2002
%t k = 8; 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 (PARI) A001244(n)=8^(n+8-1)+sum(i=1,8-1,(-1)^i/i!*(8-i)^(n+8-1)*prod(j=1,i,n+8+1-j))
%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, A000514, A001243 (columns for smaller k).
%K nonn,easy
%O 8,2
%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
%E More terms from _Christian G. Bower_, May 12 2000