A173018 Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.

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, 120, 1191, 2416, 1191, 120, 1, 0, 1, 247, 4293, 15619, 15619, 4293, 247, 1, 0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 0, 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1, 0

Offset

0,8

Comments

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.

Each row of A123125 is the reverse of the corresponding row in A173018. - Michael Somos Mar 17 2011

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

[ 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

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

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 254.

See A008292 for additional references and links.

Links

Alois P. Heinz, Rows n = 0..140, flattened

J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.

J. F. Barbero G., J. Salas and E. J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv:1307.5624 [math.CO], 2013.

Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, arXiv preprint arXiv:1105.3043 [math.CO], 2011, J. Int. Seq. 14 (2011) # 11.9.5

Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.

Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.

Digital Library of Mathematical Functions, Permutations: Order Notation

FindStat - Combinatorial Statistic Finder, The number of descents of a permutation.

A. J. J. Heidrich, On the factorization of Eulerian polynomials, Journal of Number Theory, 18(2):157-168, 1984.

F. Hirzebruch, Eulerian polynomials, Münster J. of Math. 1 (2008), pp. 9-12.

P. Hitczenko and S. Janson, Weighted random staircase tableaux, arXiv:1212.5498 [math.CO], 2012.

John M. Holte, Carries, Combinatorics and an Amazing Matrix, The American Mathematical Monthly, Vol. 104, No. 2 (Feb., 1997), pp. 138-149.

Hsien-Kuei Hwang, Hua-Huai Chern and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018.

Svante Janson, Euler-Frobenius numbers and rounding, arXiv:1305.3512 [math.PR], 2013.

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

A. Losev and Y. Manin, New moduli spaces of pointed curves and pencils of flat connections, arXiv preprint arXiv:0001003 [math.AG], 2000 (p. 8). [From Tom Copeland, Oct 03 2014]

Peter Luschny, Eulerian polynomials

John F. Sallee, The middle-cut triangulations of the n-cube, 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]

Yuriy Shablya, Dmitry Kruchinin and Vladimir Kruchinin, Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application, Mathematics (2020) Vol. 8, No. 6, 962.

M. Sheppeard, Constructive motives and scattering 2013 (p. 41). [From Tom Copeland, Oct 03 2014]

Andrei K. Svinin, Somos-4 equation and related equations, arXiv:2307.05866 [math.CA], 2023. See p. 16.

Formula

E.g.f.: (y - 1)/(y - exp(x*(y - 1))). - Geoffrey Critzer, May 04 2017

T(n, k) = Sum_{j=0..k} (-1)^j*binomial(n+1, j)*(k+1-j)^n. - G. C. Greubel, Feb 25 2019

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

Example

Triangle begins:
[ 0] 1,
[ 1] 1,    0,
[ 2] 1,    1,     0,
[ 3] 1,    4,     1,      0,
[ 4] 1,   11,    11,      1,       0,
[ 5] 1,   26,    66,     26,       1,       0,
[ 6] 1,   57,   302,    302,      57,       1,      0,
[ 7] 1,  120,  1191,   2416,    1191,     120,      1,     0,
[ 8] 1,  247,  4293,  15619,   15619,    4293,    247,     1,    0,
[ 9] 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1, 0,
[10] 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1, 0.

Maple

T:= proc(n, k) option remember;
      if k=0 and n>=0 then 1
    elif k<0 or  k>n  then 0
    else (n-k) * T(n-1, k-1) + (k+1) * T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 14 2011
# Maple since version 13:
A173018 := (n, k) -> combinat[eulerian1](n, k): # Peter Luschny, Nov 11 2012
# Or:
egf := 1 + log((x*exp(-t) - exp(-t*x))/(x-1))/(t*x):
ser := series(egf, t, 12): ct := n -> coeff(ser, t, n):
seq(print(seq((-1)^n*(n+1)!*coeff(ct(n), x, k), k=0..n)), n=0..8); # Peter Luschny, Aug 12 2022

Mathematica

t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0;
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]]
(* Jean-François Alcover, Apr 29 2011, after Maple program *)
<< Combinatorica`
Flatten[Table[Eulerian[n, k], {n, 0, 20}, {k, 0, n}]]
(* To generate the table of the numbers T(n, k) *)
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 *)
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 *)

Prog

(Sage)
@CachedFunction
def eulerian1(n, k):
    if k==0: return 1
    if k==n: return 0
    return eulerian1(n-1, k)*(k+1)+eulerian1(n-1, k-1)*(n-k)
for n in (0..9): [eulerian1(n, k) for k in(0..n)] # Peter Luschny, Nov 11 2012
(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
(Haskell)
a173018 n k = a173018_tabl !! n !! k
a173018_row n = a173018_tabl !! n
a173018_tabl = map reverse a123125_tabl
-- Reinhard Zumkeller, Nov 06 2013
(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
(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
(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]] >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2020

Crossrefs

Row sums give A000142.

Cf. A008292, A119879, A131758.

See A008517 and A201637 for the second-order numbers.

Cf. A123125 (row reversed version).

For this triangle read mod m for m=2 through 10 see A290452-A290460. See also A047999 for the mod 2 version.

Sequence in context: A294490 A085852 A123125 * A369971 A362868 A055105

Adjacent sequences: A173015 A173016 A173017 * A173019 A173020 A173021

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Nov 21 2010

Status

approved