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%I #163 Jul 25 2023 10:59:06
%S 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 120,1191,2416,1191,120,1,0,1,247,4293,15619,15619,4293,247,1,0,1,502,
%U 14608,88234,156190,88234,14608,502,1,0,1,1013,47840,455192,1310354,1310354,455192,47840,1013,1,0
%N Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
%C 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 Each row of A123125 is the reverse of the corresponding row in A173018. - _Michael Somos_ Mar 17 2011
%C 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 [ 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 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 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 254.
%D See A008292 for additional references and links.
%H Alois P. Heinz, <a href="/A173018/b173018.txt">Rows n = 0..140, flattened</a>
%H 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 J. F. Barbero G., J. Salas and E. 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 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 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 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 Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/26.14#i">Permutations: Order Notation</a>
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000021">The number of descents of a permutation.</a>
%H A. 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 F. 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 P. Hitczenko and S. Janson, <a href="http://arxiv.org/abs/1212.5498">Weighted random staircase tableaux</a>, arXiv:1212.5498 [math.CO], 2012.
%H 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 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 Svante Janson, <a href="http://arxiv.org/abs/1305.3512">Euler-Frobenius numbers and rounding</a>, arXiv:1305.3512 [math.PR], 2013.
%H 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 A. Losev and Y. 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 Peter Luschny, <a href="http://www.luschny.de/math/euler/EulerianPolynomials.html">Eulerian polynomials</a>
%H 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 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 M. 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 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 E.g.f.: (y - 1)/(y - exp(x*(y - 1))). - _Geoffrey Critzer_, May 04 2017
%F T(n, k) = Sum_{j=0..k} (-1)^j*binomial(n+1, j)*(k+1-j)^n. - _G. C. Greubel_, Feb 25 2019
%F 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 Triangle begins:
%e [ 0] 1,
%e [ 1] 1, 0,
%e [ 2] 1, 1, 0,
%e [ 3] 1, 4, 1, 0,
%e [ 4] 1, 11, 11, 1, 0,
%e [ 5] 1, 26, 66, 26, 1, 0,
%e [ 6] 1, 57, 302, 302, 57, 1, 0,
%e [ 7] 1, 120, 1191, 2416, 1191, 120, 1, 0,
%e [ 8] 1, 247, 4293, 15619, 15619, 4293, 247, 1, 0,
%e [ 9] 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 0,
%e [10] 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1, 0.
%p T:= proc(n, k) option remember;
%p if k=0 and n>=0 then 1
%p elif k<0 or k>n then 0
%p else (n-k) * T(n-1, k-1) + (k+1) * T(n-1, k)
%p fi
%p end:
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Jan 14 2011
%p # Maple since version 13:
%p A173018 := (n,k) -> combinat[eulerian1](n,k): # _Peter Luschny_, Nov 11 2012
%p # Or:
%p egf := 1 + log((x*exp(-t) - exp(-t*x))/(x-1))/(t*x):
%p ser := series(egf, t, 12): ct := n -> coeff(ser, t, n):
%p seq(print(seq((-1)^n*(n+1)!*coeff(ct(n), x, k), k=0..n)), n=0..8); # _Peter Luschny_, Aug 12 2022
%t t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0;
%t 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 (* _Jean-François Alcover_, Apr 29 2011, after Maple program *)
%t << Combinatorica`
%t Flatten[Table[Eulerian[n, k], {n, 0, 20}, {k, 0, n}]]
%t (* To generate the table of the numbers T(n,k) *)
%t 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 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 (Sage)
%o @CachedFunction
%o def eulerian1(n, k):
%o if k==0: return 1
%o if k==n: return 0
%o return eulerian1(n-1, k)*(k+1)+eulerian1(n-1, k-1)*(n-k)
%o for n in (0..9): [eulerian1(n, k) for k in(0..n)] # _Peter Luschny_, Nov 11 2012
%o (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 (Haskell)
%o a173018 n k = a173018_tabl !! n !! k
%o a173018_row n = a173018_tabl !! n
%o a173018_tabl = map reverse a123125_tabl
%o -- _Reinhard Zumkeller_, Nov 06 2013
%o (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 (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
%o (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 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 28 2020
%Y Row sums give A000142.
%Y Cf. A008292, A119879, A131758.
%Y See A008517 and A201637 for the second-order numbers.
%Y Cf. A123125 (row reversed version).
%Y For this triangle read mod m for m=2 through 10 see A290452-A290460. See also A047999 for the mod 2 version.
%K nonn,tabl,easy
%O 0,8
%A _N. J. A. Sloane_, Nov 21 2010
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