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 A173018 #171 Aug 22 2025 10:57:24
%S A173018 1,1,0,1,1,0,1,4,1,0,1,11,11,1,0,1,26,66,26,1,0,1,57,302,302,57,1,0,1,
%T A173018 120,1191,2416,1191,120,1,0,1,247,4293,15619,15619,4293,247,1,0,1,502,
%U A173018 14608,88234,156190,88234,14608,502,1,0,1,1013,47840,455192,1310354,1310354,455192,47840,1013,1,0
%N A173018 Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
%C A173018 This version indexes the Eulerian numbers in the same way as Graham et al.'s Concrete Mathematics (see references section). The traditional indexing, used by Riordan, Comtet and others, is given in A008292, which is the main entry for the Eulerian numbers.
%C A173018 Each row of A123125 is the reverse of the corresponding row in A173018. - _Michael Somos_ Mar 17 2011
%C A173018 Triangle T(n,k), read by rows, given by [1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...] DELTA [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_ Sep 30 2011
%C A173018 [ E(.,t)/(1-t)]^n = n!*Lag[n,-P(.,t)/(1-t)] and [ -P(.,t)/(1-t)]^n = n!*Lag[n, E(.,t)/(1-t)] umbrally comprise a combinatorial Laguerre transform pair, where E(n,t) are the Eulerian polynomials (e.g., E(2,t)= 1+t) and P(n,t) are the polynomials related to polylogarithms in A131758. - _Tom Copeland_, Oct 03 2014
%C A173018 See A131758 for connections of the evaluation of these polynomials at -1 (alternating row sum) to the Euler, Genocchi, Bernoulli, and zag/tangent numbers and values of the Riemann zeta function and polylogarithms. See also A119879 for the Swiss-knife polynomials. - _Tom Copeland_, Oct 20 2015
%D A173018 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 254.
%D A173018 See A008292 for additional references and links.
%H A173018 Alois P. Heinz, <a href="/A173018/b173018.txt">Rows n = 0..140, flattened</a>
%H A173018 J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013.
%H A173018 J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.5624">Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications</a>, arXiv:1307.5624 [math.CO], 2013.
%H A173018 Paul Barry, <a href="http://arxiv.org/abs/1105.3043">Eulerian polynomials as moments, via exponential Riordan arrays</a>, arXiv preprint arXiv:1105.3043 [math.CO], 2011, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry7/barry172.html">J. Int. Seq. 14 (2011) # 11.9.5</a>
%H A173018 Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.
%H A173018 Paul Barry, <a href="https://arxiv.org/abs/1803.10297">Generalized Eulerian Triangles and Some Special Production Matrices</a>, arXiv:1803.10297 [math.CO], 2018.
%H A173018 Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/26.14#i">Permutations: Order Notation</a>
%H A173018 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000021">The number of descents of a permutation.</a>
%H A173018 Arnošt J. J. Heidrich, <a href="https://doi.org/10.1016/0022-314X(84)90050-7">On the factorization of Eulerian polynomials</a>, Journal of Number Theory, 18(2):157-168, 1984.
%H A173018 Friedrich Hirzebruch, <a href="https://www.uni-muenster.de/FB10/mjm/vol_1/mjm_vol_1_02.pdf">Eulerian polynomials</a>, Münster J. of Math. 1 (2008), pp. 9-12.
%H A173018 Pawel Hitczenko and Svante Janson, <a href="http://arxiv.org/abs/1212.5498">Weighted random staircase tableaux</a>, arXiv:1212.5498 [math.CO], 2012.
%H A173018 John M. Holte, <a href="http://www.jstor.org/stable/2974981">Carries, Combinatorics and an Amazing Matrix</a>, The American Mathematical Monthly, Vol. 104, No. 2 (Feb., 1997), pp. 138-149.
%H A173018 Hsien-Kuei Hwang, Hua-Huai Chern and Guan-Huei Duh, <a href="https://arxiv.org/abs/1807.01412">An asymptotic distribution theory for Eulerian recurrences with applications</a>, arXiv:1807.01412 [math.CO], 2018.
%H A173018 Svante Janson, <a href="http://arxiv.org/abs/1305.3512">Euler-Frobenius numbers and rounding</a>, arXiv:1305.3512 [math.PR], 2013.
%H A173018 Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%H A173018 Andrey Losev and Yuri Manin, <a href="http://arxiv.org/abs/math/0001003">New moduli spaces of pointed curves and pencils of flat connections</a>, arXiv preprint arXiv:0001003 [math.AG], 2000 (p. 8). [From _Tom Copeland_, Oct 03 2014]
%H A173018 Peter Luschny, <a href="http://www.luschny.de/math/euler/EulerianPolynomials.html">Eulerian polynomials</a>
%H A173018 John F. Sallee, <a href="http://dx.doi.org/10.1137/0605039">The middle-cut triangulations of the n-cube</a>, SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 407--419. MR0752044 (86c:05054). See Table 1. [From _N. J. A. Sloane_, Apr 09 2014]
%H A173018 Yuriy Shablya, Dmitry Kruchinin and Vladimir Kruchinin, <a href="https://doi.org/10.3390/math8060962">Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application</a>, Mathematics (2020) Vol. 8, No. 6, 962.
%H A173018 Marni Dee Sheppeard, <a href="http://vixra.org/pdf/1208.0242v6.pdf">Constructive motives and scattering</a> 2013 (p. 41). [From _Tom Copeland_, Oct 03 2014]
%H A173018 Andrei K. Svinin, <a href="https://arxiv.org/abs/2307.05866">Somos-4 equation and related equations</a>, arXiv:2307.05866 [math.CA], 2023. See p. 16.
%F A173018 E.g.f.: (y - 1)/(y - exp(x*(y - 1))). - _Geoffrey Critzer_, May 04 2017
%F A173018 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(n+1, j)*(k+1-j)^n. - _G. C. Greubel_, Feb 25 2019
%F A173018 T(n, k) = (-1)^n*(n+1)!*[x^k][t^n](1 + log((x*exp(-t) - exp(-t*x))/(x-1))/(t*x)). - _Peter Luschny_, Aug 12 2022
%e A173018 Triangle begins:
%e A173018 [ 0] 1,
%e A173018 [ 1] 1, 0,
%e A173018 [ 2] 1, 1, 0,
%e A173018 [ 3] 1, 4, 1, 0,
%e A173018 [ 4] 1, 11, 11, 1, 0,
%e A173018 [ 5] 1, 26, 66, 26, 1, 0,
%e A173018 [ 6] 1, 57, 302, 302, 57, 1, 0,
%e A173018 [ 7] 1, 120, 1191, 2416, 1191, 120, 1, 0,
%e A173018 [ 8] 1, 247, 4293, 15619, 15619, 4293, 247, 1, 0,
%e A173018 [ 9] 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 0,
%e A173018 [10] 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1, 0.
%p A173018 T:= proc(n, k) option remember;
%p A173018 if k=0 and n>=0 then 1
%p A173018 elif k<0 or k>n then 0
%p A173018 else (n-k) * T(n-1, k-1) + (k+1) * T(n-1, k)
%p A173018 fi
%p A173018 end:
%p A173018 seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Jan 14 2011
%p A173018 # Maple since version 13:
%p A173018 A173018 := (n,k) -> combinat[eulerian1](n,k): # _Peter Luschny_, Nov 11 2012
%p A173018 # Or:
%p A173018 egf := 1 + log((x*exp(-t) - exp(-t*x))/(x-1))/(t*x):
%p A173018 ser := series(egf, t, 12): ct := n -> coeff(ser, t, n):
%p A173018 seq(print(seq((-1)^n*(n+1)!*coeff(ct(n), x, k), k=0..n)), n=0..8); # _Peter Luschny_, Aug 12 2022
%t A173018 t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0;
%t A173018 t[n_,k_] := t[n,k] = (n-k)*t[n-1,k-1] + (k+1)*t[n-1, k]; Flatten[Table[t[n,k], {n,0,11}, {k,0,n}]][[1 ;; 60]]
%t A173018 (* _Jean-François Alcover_, Apr 29 2011, after Maple program *)
%t A173018 << Combinatorica`
%t A173018 Flatten[Table[Eulerian[n, k], {n, 0, 20}, {k, 0, n}]]
%t A173018 (* To generate the table of the numbers T(n,k) *)
%t A173018 RecurrenceTable[{T[n + 1, k + 1] == (n - k) T[n, k] + (k + 2) T[n, k + 1], T[0, k] == KroneckerDelta[k]}, T, {n, 0, 12}, {k, 0, 12}] (* _Emanuele Munarini_, Jan 03 2018 *)
%t A173018 Table[If[n==0,1, Sum[(-1)^j*Binomial[n+1, j]*(k+1-j)^n, {j,0,k+1}]], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 25 2019 *)
%o A173018 (Sage)
%o A173018 @CachedFunction
%o A173018 def eulerian1(n, k):
%o A173018 if k==0: return 1
%o A173018 if k==n: return 0
%o A173018 return eulerian1(n-1, k)*(k+1)+eulerian1(n-1, k-1)*(n-k)
%o A173018 for n in (0..9): [eulerian1(n, k) for k in(0..n)] # _Peter Luschny_, Nov 11 2012
%o A173018 (Sage) [1] + [[sum((-1)^(k-j+1)*binomial(n+1,k-j+1)*j^n for j in (0..k+1)) for k in (0..n)] for n in (1..12)] # _G. C. Greubel_, Feb 25 2019
%o A173018 (Haskell)
%o A173018 a173018 n k = a173018_tabl !! n !! k
%o A173018 a173018_row n = a173018_tabl !! n
%o A173018 a173018_tabl = map reverse a123125_tabl
%o A173018 -- _Reinhard Zumkeller_, Nov 06 2013
%o A173018 (Magma) [[n le 0 select 1 else (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Feb 25 2019
%o A173018 (Magma) T:= func< n,k | n eq 0 select 1 else &+[(-1)^(k-j+1)*Binomial(n+1,k-j+1)*j^n: j in [0..k+1]] >;
%o A173018 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 28 2020
%o A173018 (PARI) T(n,k) = if(n==0, 1, sum(j=0,k+1, (-1)^(k-j+1)*binomial(n+1,k-j+1)*j^n)); \\ _G. C. Greubel_, Feb 28 2020
%Y A173018 Row sums give A000142.
%Y A173018 Cf. A008292, A119879, A131758.
%Y A173018 See A008517 and A201637 for the second-order numbers.
%Y A173018 Cf. A123125 (row reversed version).
%Y A173018 For this triangle read mod m for m=2 through 10 see A290452-A290460. See also A047999 for the mod 2 version.
%K A173018 nonn,tabl,easy,changed
%O A173018 0,8
%A A173018 _N. J. A. Sloane_, Nov 21 2010