A122704 -id:A122704 - cp's OEIS frontend

A123125 Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.

1, 0, 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

Offset: 0

Philippe Deléham, Sep 30 2006

The beginning of this sequence does not quite agree with the usual version, which is A173018. - N. J. A. Sloane, Nov 21 2010
Each row of A123125 is the reverse of the corresponding row in A173018. - Michael Somos, Mar 17 2011
A008292 (subtriangle for k>=1 and n>=1) is the main entry for these numbers.
Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,...] where DELTA is the operator defined in A084938.
Row sums are the factorials. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008
If the initial zero column is deleted, the result is A008292. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008
This result gives an alternative method of calculating the Eulerian numbers by an Umbral Calculus expansion from Comtet. - Roger L. Bagula, Nov 21 2009
This function seems to be equivalent to the PolyLog expansion. - Roger L. Bagula, Nov 21 2009
A raising operator formed from the e.g.f. of this entry is the generator of a sequence of polynomials p(n,x;t) defined in A046802 that specialize to those for A119879 as p(n,x;-1), A007318 as p(n,x;0), A073107 as p(n,x;1), and A046802 as p(n,0;t). See Copeland link for more associations. - Tom Copeland, Oct 20 2015
The Eulerian numbers in this setup count the permutation trees of power n and width k (see the Luschny link). For the associated combinatorial statistic over permutations see the Sage program below and the example section. - Peter Luschny, Dec 09 2015 [See Elder et al. link. Peter Luschny, Jul 13 2022]
From Wolfdieter Lang, Apr 03 2017: (Start)
The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k are the numerator polynomials of the o.g.f. G(n, x) of n-powers {m^n}_{m>=0} (with 0^0 = 1): G(n, x) = R(n, x)/(1-x)^(n+1). See the Aug 14 2008 formula, where f(x,n) = R(n, x). The e.g.f. of R(n, t) is given in Copeland's Oct 14 2015 formula below.
The first nine column sequences are A000007, A000012, A000295, A000460, A000498, A000505, A000514, A001243, A001244. (End)
With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of this entry, A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020
Let b(n) = (1/(n+1))*Sum_{k=0..n-1} (-1)^(n-k+1)*T(n, k+1) / binomial(n, k+1). Then b(n) = Bernoulli(n, 1) = -n*Zeta(1 - n) = Integral_{x=0..1} F_n(x) for n >= 1. Here F_n(x) are the signed Fubini polynomials (A278075). (See also Rzadkowski and Urlinska, example 1.) - Peter Luschny, Feb 15 2021
Patrick J. Burchell (see link) describes the following method: To get the k-th row of the triangle write the nonnegative integers with a fixed exponent k as a sequence, 0^k, 1^k, 2^k, ..., and then apply the first differences to them k + 1 times. - Peter Luschny, Apr 02 2023

			The triangle T(n, k) begins:
  n\k 0 1    2     3      4       5       6      7     8    9 10...
  0:  1
  1:  0 1
  2:  0 1    1
  3:  0 1    4     1
  4:  0 1   11    11      1
  5:  0 1   26    66     26       1
  6:  0 1   57   302    302      57       1
  7:  0 1  120  1191   2416    1191     120      1
  8:  0 1  247  4293  15619   15619    4293    247     1
  9:  0 1  502 14608  88234  156190   88234  14608   502    1
 10:  0 1 1013 47840 455192 1310354 1310354 455192 47840 1013  1
...  Reformatted. - _Wolfdieter Lang_, Feb 14 2015
------------------------------------------------------------------
The width statistic over permutations, n=4.
  [1, 2, 3, 4] => 3; [1, 2, 4, 3] => 2; [1, 3, 2, 4] => 2; [1, 3, 4, 2] => 2;
  [1, 4, 2, 3] => 2; [1, 4, 3, 2] => 1; [2, 1, 3, 4] => 3; [2, 1, 4, 3] => 2;
  [2, 3, 1, 4] => 2; [2, 3, 4, 1] => 3; [2, 4, 1, 3] => 2; [2, 4, 3, 1] => 2;
  [3, 1, 2, 4] => 3; [3, 1, 4, 2] => 3; [3, 2, 1, 4] => 2; [3, 2, 4, 1] => 3;
  [3, 4, 1, 2] => 3; [3, 4, 2, 1] => 2; [4, 1, 2, 3] => 4; [4, 1, 3, 2] => 3;
  [4, 2, 1, 3] => 3; [4, 2, 3, 1] => 3; [4, 3, 1, 2] => 3; [4, 3, 2, 1] => 2;
Gives row(4) = [0, 1, 11, 11, 1]. - _Peter Luschny_, Dec 09 2015
------------------------------------------------------------------
From _Wolfdieter Lang_, Apr 03 2017: (Start)
Recurrence: T(5, 3) = (6-3)*T(4, 2) + 3*T(4, 3) = 3*11 + 3*11 = 66.
O.g.f. column k=2: (x/(1 - 2*x))*E_x*(x/(1-x)) = (x/(1-x))^2/(1-2*x).
E.g.f. column k=2: A(2, x) = x*A(1, x) + x*E(1, x) = x*1 + x*(exp(x)-1) = x*exp(x), hence E(2, x) = (1 + int(x*exp(-x),x ))*exp(2*x) = exp(x)*(exp(x) - (1+x)). See A000295. (End)

L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245. [Roger L. Bagula, Nov 21 2009]
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed.; Addison-Wesley, 1994, p. 268, Row reversed table 268. - Wolfdieter Lang, Apr 03 2017
Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
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, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.
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.
V. Batyrev and M. Blume, The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces, p. 11, arXiv:/0911.3607 [math.AG], 2009. [_Tom Copeland_, Oct 16 2015]
Anna Borowiec and Wojciech Mlotkowski, New Eulerian numbers of type D, arXiv:1509.03758 [math.CO], 2015.
Patrick J. Burchell, A Generalisation of Ramanujan's (back of the envelope) Method for Divergent Series, arXiv:2303.14045 math.NT, 2023.
A. Cohen, Eulerian polynomials of spherical type, Münster J. of Math. 1 (2008). [_Tom Copeland_, Oct 16 2015]
Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera
Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker and Amanda Welch, Homomesies on permutations -- an analysis of maps and statistics in the FindStat database, arXiv:2206.13409 [math.CO], 2022-2023. (Def. 4.20 and Prop. 4.22.)
FindStat - Combinatorial Statistic Finder, The number of descents of a permutation.
F. Hirzebruch, Eulerian polynomials, Münster J. of Math. 1 (2008), pp. 9-12.
P. Hitczenko and S. Janson, Weighted random staircase tableaux, arXiv preprint arXiv:1212.5498 [math.CO], 2012.
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 preprint arXiv:1305.3512 [math.PR], 2013.
Katarzyna Kril and Wojciech Mlotkowski, Permutations of Type B with Fixed Number of Descents and Minus Signs, Volume 26(1) of The Electronic Journal of Combinatorics, 2019.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 22.
A. Losev and Y. Manin, New moduli spaces of pointed curves and pencils of flat connections, arXiv preprint arXiv:math/0001003 [math.AG], 2000 (p. 8). - _Tom Copeland_, Oct 16 2015
Peter Luschny, Permutation Trees
G. Rzadkowski and M. Urlinska, A Generalization of the Eulerian Numbers, arXiv:1612.06635 [math.CO], 2016.

See A008292 (subtriangle for k>=1 and n>=1), which is the main entry for these numbers. Another version has the zeros at the ends of the rows, as in Concrete Mathematics: see A173018.
Cf. A007318, A130850, A028246, A046802, A009006, A019538, A090582, A119879, A248727.
T(2n,n) gives A180056.

a123125 n k = a123125_tabl !! n !! k
a123125_row n = a123125_tabl !! n
a123125_tabl = [1] : zipWith (:) [0, 0 ..] a008292_tabl
-- Reinhard Zumkeller, Nov 06 2013

gf := 1/(1 - t*exp(x)): ser := series(gf, x, 12):
cx := n -> (-1)^(n + 1)*factor(n!*coeff(ser, x, n)*(t - 1)^(n + 1)):
seq(print(seq(coeff(cx(n), t, k), k = 0..n)), n = 0..9); # Peter Luschny, Feb 11 2021
A123125 := proc(n, k) option remember; if k = n then 1 elif k <= 0 or k > n then 0 else k*procname(n-1, k) + (n-k+1)*procname(n-1, k-1) fi end:
seq(print(seq(A123125(n, k), k=0..n)), n=0..10); # Peter Luschny, Mar 28 2021
# Alternative (Patrick J. Burchell):
t := a -> Statistics:-Difference([0, a]): Trow := k -> (t@@(k+1))([seq(n^k, n = 0..k)]):
seq(print(Trow(n)), n = 0..6); # Peter Luschny, Apr 02 2023

f[x_, n_] := f[x, n] = (1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}];
Table[CoefficientList[f[x, n], x], {n,0,9}] // Flatten (* Roger L. Bagula, Aug 14 2008 *)
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]; T[n_, k_] := t[n, n-k];
Table[T[n, k], {n,0,10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019 *)
A123125[n_, k_] := Sum[(-1)^j*(n - j - k + 1)^n * Binomial[n + 1, j], {j, 0, n - k}];
Table[A123125[n, k], {n, 0, 9}, {k, 0, n}] // TableForm  (* Peter Luschny, Aug 12 2022 *)

from math import isqrt, comb
def A123125(n):
    a = (m:=isqrt(k:=n+1<<1))+(k>m*(m+1))
    b = comb(a+1,2)-n
    return sum(-(b-j)**(a-1)*comb(a,j) if j&1 else (b-j)**(a-1)*comb(a,j) for j in range(b)) # Chai Wah Wu, Nov 13 2024

def statistic_eulerian(pi):
    if not pi: return 0
    h, i, branch, next = 0, len(pi), [0], pi[0]
    while True:
        while next < branch[len(branch)-1]:
            del(branch[len(branch)-1])
        current = 0
        h += 1
        while next > current:
            i -= 1
            if i == 0: return h
            branch.append(next)
            current, next = next, pi[i]
def A123125_row(n):
    L = [0]*(n+1)
    for p in Permutations(n):
        L[statistic_eulerian(p)] += 1
    return L
[A123125_row(n) for n in range(7)] # Peter Luschny, Dec 09 2015

Sum_{k=0..n} T(n,k) = n! = A000142(n).
Sum_{k=0..n} 2^k*T(n,k) = A000629(n).
Sum_{k=0..n} 3^k*T(n,k) = abs(A009362(n+1)).
Sum_{k=0..n} 2^(n-k)*T(n,k) = A000670(n).
Sum_{k=0..n} T(n,k)*3^(n-k) = A122704(n). - Philippe Deléham, Nov 07 2007
G.f.: f(x,n) = (1 - x)^(n + 1)*Sum_{k>=0} k^n*x^k. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008. f is not the g.f. of the triangle, it is the polynomial of row n. See an Apr 03 2017 comment above - Wolfdieter Lang, Apr 03 2017
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000629(n), A123227(n), A201355(n), A201368(n) for x = 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Dec 01 2011
E.g.f. (1-t)/(1-t*exp((1-t)x)). A123125 * A007318 = A130850 = unsigned A075263, related to reversed A028246. A007318 * A123125 = A046802. Evaluating the row polynomials at -1, giving the alternating-sign row sum, generates A009006. - Tom Copeland, Oct 14 2015
From Wolfdieter Lang, Apr 03 2017: (Start)
T(n, k) = A173018(n, n-k), 0 <= k <= n. Row reversed Euler's triangle. See Graham et al., p. 268.
Recurrence (from A173018): T(n, 0) = 1 if n=0 else 0; T(n, k) = 0 if n < k and T(n, k) = (n+1-k)*T(n-1, k-1) + k*T(n-1, k) else.
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n-j, k-j)*S2(n, j)*j!, 0 <= k <= n, else 0. For S2(n, k)*k! see A131689.
The recurrence for the o.g.f. of the sequence of column k is
  G(k, x) = (x/(1 - k*x))*(E_x - (k-2))*G(k-1, x), with the Euler operator E_x  = x*d_x, for k >= 1, with G(0, x) = 1. (Proof from the recurrence of T(n, k)).
The e.g.f of the sequence of column k is found from E(k, x) = (1 + int(A(k, x),x)*exp(-k*x))*exp(k*x), k >= 1, with the recurrence
  A(k, x) = x*A(k-1, x) +(1 + (1-k)*(1-x))*E(k-1, x) for k >= 1, with  A(0,x)= 0. (Proof from the recurrence of T(n, k)). (End)
T(n, k) = Sum_{j=0..n-k} (-1)^j*(n-j-k+1)^n*binomial(n + 1, j). - Peter Luschny, Aug 12 2022
G.f.: Sum_{m >= 0} x^m/(1/(1-x)-m*t). - Mamuka Jibladze, Mar 12 2025

A131689 Triangle of numbers T(n,k) = k!Stirling2(n,k) = A000142(k)A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880

Offset: 0

Philippe Deléham, Sep 14 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is the operator defined in A084938; another version of A019538.
See also A019538: version with n > 0 and k > 0. - Philippe Deléham, Nov 03 2008
From Peter Bala, Jul 21 2014: (Start)
T(n,k) gives the number of (k-1)-dimensional faces in the interior of the first barycentric subdivision of the standard (n-1)-dimensional simplex. For example, the barycentric subdivision of the 1-simplex is o--o--o, with 1 interior vertex and 2 interior edges, giving T(2,1) = 1 and T(2,2) = 2.
This triangle is used when calculating the face vectors of the barycentric subdivision of a simplicial complex. Let S be an n-dimensional simplicial complex and write f_k for the number of k-dimensional faces of S, with the usual convention that f_(-1) = 1, so that F := (f_(-1), f_0, f_1,...,f_n) is the f-vector of S. If M(n) denotes the square matrix formed from the first n+1 rows and n+1 columns of the present triangle, then the vector F*M(n) is the f-vector of the first barycentric subdivision of the simplicial complex S (Brenti and Welker, Lemma 2.1). For example, the rows of Pascal's triangle A007318 (but with row and column indexing starting at -1) are the f-vectors for the standard n-simplexes. It follows that A007318*A131689, which equals A028246, is the array of f-vectors of the first barycentric subdivision of standard n-simplexes. (End)
This triangle T(n, k) appears in the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} (x^k/(1 - x)^(k+2))*T(n, k). See also the Eulerian triangle A008292 with a Mar 31 2017 comment for a rewritten form. For the e.g.f. see A028246 with a Mar 13 2017 comment. - Wolfdieter Lang, Mar 31 2017
T(n,k) = the number of alignments of length k of n strings each of length 1. See Slowinski. An example is given below. Cf. A122193 (alignments of strings of length 2) and A299041 (alignments of strings of length 3). - Peter Bala, Feb 04 2018
The row polynomials R(n,x) are the Fubini polynomials. - Emanuele Munarini, Dec 05 2020
From Gus Wiseman, Feb 18 2022: (Start)
Also the number of patterns of length n with k distinct parts (or with maximum part k), where we define a pattern to be a finite sequence covering an initial interval of positive integers. For example, row n = 3 counts the following patterns:
  (1,1,1)  (1,2,2)  (1,2,3)
           (2,1,2)  (1,3,2)
           (2,2,1)  (2,1,3)
           (1,1,2)  (2,3,1)
           (1,2,1)  (3,1,2)
           (2,1,1)  (3,2,1)
(End)
Regard A048994 as a lower-triangular matrix and divide each term A048994(n,k) by n!, then this is the matrix inverse. Because Sum_{k=0..n} (A048994(n,k) * x^n / n!) = A007318(x,n), Sum_{k=0..n} (A131689(n,k) * A007318(x,k)) = x^n. - Natalia L. Skirrow, Mar 23 2023
T(n,k) is the number of ordered partitions of [n] into k blocks. - Alois P. Heinz, Feb 21 2025

			The triangle T(n,k) begins:
  n\k 0 1    2     3      4       5        6        7        8        9      10 ...
  0:  1
  1:  0 1
  2:  0 1    2
  3:  0 1    6     6
  4:  0 1   14    36     24
  5:  0 1   30   150    240     120
  6:  0 1   62   540   1560    1800      720
  7:  0 1  126  1806   8400   16800    15120     5040
  8:  0 1  254  5796  40824  126000   191520   141120    40320
  9:  0 1  510 18150 186480  834120  1905120  2328480  1451520   362880
  10: 0 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800
  ... reformatted and extended. - _Wolfdieter Lang_, Mar 31 2017
From _Peter Bala_, Feb 04 2018: (Start)
T(4,2) = 14 alignments of length 2 of 4 strings of length 1. Examples include
  (i) A -    (ii) A -    (iii) A -
      B -         B -          - B
      C -         - C          - C
      - D         - D          - D
There are C(4,1) = 4 alignments of type (i) with a single gap character - in column 1, C(4,2) = 6 alignments of type (ii) with two gap characters in column 1 and C(4,3) = 4 alignments of type (iii) with three gap characters in column 1, giving a total of 4 + 6 + 4 = 14 alignments. (End)

Vincenzo Librandi, Rows n = 0..100, flattened
Peter Bala, Deformations of the Hadamard product of power series
F. Brenti and V. Welker, f-vectors of barycentric subdivisions, arXiv:math/0606356 [math.CO], Math. Z., 259(4), 849-865, 2008.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Massimo Nocentini, An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation, PhD Thesis, University of Florence, 2019. See Ex. 36.
Mircea Dan Rus, Yet another note on notation, arXiv:2501.08762 [math.HO], 2025. See p. 6.
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
Wikipedia, Barycentric subdivision
Wikipedia, Simplicial complex
Wikipedia, Simplex
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Columns k=0..10 are A000007, A000012, A000918, A001117, A000919, A001118, A000920, A135456, A133068, A133360, A133132,
Case m=1 of the polynomials defined in A278073.
Cf. A000142 (diagonal), A000670 (row sums), A000012 (alternating row sums), A210029 (central terms).
Cf. A001286, A037960, A037961, A037962, A037963.
Cf. A008292, A028246 (o.g.f. and e.g.f. of sums of powers).
Cf. A019538, A122193, A299041.
A version for partitions is A116608, or by maximum A008284.
A version for compositions is A235998, or by maximum A048004.
Classes of patterns:
- A000142 = strict
- A005649 = anti-run, complement A069321
- A019536 = necklace
- A032011 = distinct multiplicities
- A060223 = Lyndon
- A226316 = (1,2,3)-avoiding, weakly A052709, complement A335515
- A296975 = aperiodic
- A345194 = alternating, up/down A350354, complement A350252
- A349058 = weakly alternating
- A351200 = distinct runs
- A351292 = distinct run-lengths
Cf. A007318, A097805, A335456, A335457, A344605, A349050.

function T(n, k)
    if k < 0 || k > n return 0 end
    if n == 0 && k == 0 return 1 end
    k*(T(n-1, k-1) + T(n-1, k))
end
for n in 0:7
    println([T(n, k) for k in 0:n])
end
# Peter Luschny, Mar 26 2020

A131689 := (n,k) -> Stirling2(n,k)*k!: # Peter Luschny, Sep 17 2011
# Alternatively:
A131689_row := proc(n) 1/(1-t*(exp(x)-1)); expand(series(%,x,n+1)); n!*coeff(%,x,n); PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 9 do A131689_row(n) od; # Peter Luschny, Jan 23 2017

t[n_, k_] := k!*StirlingS2[n, k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014 *)
T[n_, k_] := If[n <= 0 || k <= 0, Boole[n == 0 && k == 0], Sum[(-1)^(i + k) Binomial[k, i] i^(n + k), {i, 0, k}]]; (* Michael Somos, Jul 08 2018 *)

{T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(k + i) * binomial(k, i) * i^n))};
/* Michael Somos, Jul 08 2018 */

@cached_function
def F(n): # Fubini polynomial
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
for n in (0..9): print(F(n).list()) # Peter Luschny, May 21 2021

T(n,k) = k*(T(n-1,k-1) + T(n-1,k)) with T(0,0)=1. Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A000629(n), A033999(n), A000007(n), A000670(n), A004123(n+1), A032033(n), A094417(n), A094418(n), A094419(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively. [corrected by Philippe Deléham, Feb 11 2013]
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A000670(n), A122704(n) for x=-1, 0, 1, 2 respectively. - Philippe Deléham, Oct 09 2007
Sum_{k=0..n} (-1)^k*T(n,k)/(k+1) = Bernoulli numbers A027641(n)/A027642(n). - Peter Luschny, Sep 17 2011
G.f.: F(x,t) = 1 + x*t + (x+x^2)*t^2/2! + (x+6*x^2+6*x^3)*t^3/3! + ... = Sum_{n>=0} R(n,x)*t^n/n!.
The row polynomials R(n,x) satisfy the recursion R(n+1,x) = (x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x. - Philippe Deléham, Feb 11 2013
T(n,k) = [t^k] (n! [x^n] (1/(1-t*(exp(x)-1)))). - Peter Luschny, Jan 23 2017
The n-th row polynomial has the form x o x o ... o x (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See also Bala, Example E8. - Peter Bala, Jan 08 2018

A336950 E.g.f.: 1 / (1 - x * exp(2*x)).

Original entry on oeis.org

1, 1, 6, 42, 392, 4600, 64752, 1063216, 19952256, 421227648, 9880951040, 254960721664, 7176891675648, 218857588139008, 7187394935347200, 252897556424140800, 9491754142468702208, 378509920569294684160, 15982018774576565649408, 712306819507400060502016

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..394

Column k=2 of A351790.

Cf. A001787, A006153, A122704, A216794, A235328, A336947, A336951, A336952.

nmax = 19; CoefficientList[Series[1/(1 - x Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(2 (n - k))^k/k!, {k, 0, n}], {n, 1, 19}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

seq(n)={ Vec(serlaplace(1 / (1 - x*exp(2*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (2 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 2^(k-1) * a(n-k).
a(n) ~ n! * (2/LambertW(2))^n / (1 + LambertW(2)). - Vaclav Kotesovec, Aug 09 2021

A255927 a(n) = (3/4) * Sum_{k>=0} (3*k)^n/4^k.

Original entry on oeis.org

1, 1, 5, 33, 285, 3081, 40005, 606033, 10491885, 204343641, 4422082005, 105265315233, 2733583519485, 76902684021801, 2329889536156005, 75629701786875633, 2618654297178083085, 96336948993312237561, 3752590641305604502005, 154294551397830418471233, 6677999524135208461382685

Offset: 0

Karol A. Penson, Sep 03 2015

nonn

			a(5) = 729*hypergeom([2,2,2,2,2],[1,1,1,1],1/4)/16 = 3081.

Alois P. Heinz, Table of n, a(n) for n = 0..400
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the Log-normal distribution, arXiv:quant-ph/0303030, 2003.
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the Log-normal distribution, J. Phys. A: Math. Gen. 36, (2003), L273.
Eric Weisstein's World of Mathematics, Lerch Transcendent

Cf. A000670, A094418, A004123 A032033, A094417, A122704, A326323.

S:= series(3/(4-exp(3*x)), x, 51):
seq(coeff(S,x,n)*n!, n=0..50); # Robert Israel, Sep 03 2015
seq(add(combinat:-eulerian1(n,k)*4^k, k=0..n), n=0..20); # Peter Luschny, Jun 27 2019

a[n_] := 3^(n+1)/4 HurwitzLerchPhi[1/4, -n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 18 2018 *)
Eulerian1[0, 0] = 1; Eulerian1[n_, k_] := Sum[(-1)^j (k-j+1)^n Binomial[n+1, j], {j, 0, k+1}]; Table[Sum[Eulerian1[n, k] 4^k, {k, 0, n}], {n, 0, 20}] (* Jean-François Alcover, Jul 13 2019, after Peter Luschny *)

a(n) = sum(k=0, n, stirling(n,k,2)*k!*3^(n-k)); \\ Michel Marcus, Sep 03 2015

a(n) = Sum_{k>=0} Stirling2(n,k)*k!*3^(n-k).
E.g.f.: 3/(4-exp(3*x)).
Special values of the generalized hypergeometric function n_F_(n-1):
a(n) = (3^(n+1)/16) * hypergeom([2,2,..2],[1,1,..1],1/4), where the sequence in the first square bracket ("upper" parameters) has n elements all equal to 2 whereas the sequence in the second square bracket ("lower" parameters) has n-1 elements all equal to 1.
Example: a(5) = 729 * hypergeom([2,2,2,2,2],[1,1,1,1],1/4)/16 = 3081.
a(n) is the n-th moment of the discrete weight function W(x) = (3/4)*sum(k>=0, Dirac(x-3*k)/4^k), n>=1.
a(n) ~ n! * 3^(n+1) / ((log(2))^(n+1) * 2^(n+3)). - Vaclav Kotesovec, Jul 09 2018
G.f.: Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - 3*k*x). - Ilya Gutkovskiy, Apr 04 2019
a(n) = A_{4}(n) where A_{n}(x) are the Eulerian polynomials as defined in A326323. - Peter Luschny, Jun 27 2019

a(0)=1 prepended by Alois P. Heinz, Sep 18 2018

A326324 a(n) = A_{5}(n) where A_{m}(x) are the Eulerian polynomials as defined in A326323.

Original entry on oeis.org

1, 1, 6, 46, 456, 5656, 84336, 1467376, 29175936, 652606336, 16219458816, 443419545856, 13224580002816, 427278468668416, 14867050125981696, 554245056343668736, 22039796215883268096, 931198483176870608896, 41658202699736550014976, 1967160945260218035798016

Offset: 0

Peter Luschny, Jun 27 2019

nonn

See A326323 for the more general formulas.

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326323.

seq(add(combinat:-eulerian1(n,k)*5^k, k=0..n), n=0..20);
# Alternative:
egf := 4/(5 - exp(4*x)): ser := series(egf, x, 21):
seq(k!*coeff(ser, x, k), k=0..20);

a[1] := 1; a[n_] := 4^(n + 1)/5 HurwitzLerchPhi[1/5, -n, 0];
Table[a[n], {n, 0, 20}]
(* Alternative: *)
s[n_] := Sum[StirlingS2[n, j] 4^(n - j) j!, {j, 0, n}];
Table[s[n], {n, 0, 20}]

a(n) ~ n!/5 * (4/log(5))^(n+1). - Vaclav Kotesovec, Aug 09 2021

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k). - Ilya Gutkovskiy, Feb 04 2022

Corrected after notice from Jean-François Alcover by Peter Luschny, Jul 13 2019

A352860 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k) * 2^k * a(k).

Original entry on oeis.org

1, 1, 5, 67, 2273, 187411, 36539465, 16496912587, 16958655627233, 39148957534778851, 200638280176080172025, 2261092739579072893806907, 55582179517311967755693514193, 2960001149710485505367113202321491, 339497331023047752386812273780566932585

Offset: 0

Ilya Gutkovskiy, Apr 06 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..77

Cf. A000670, A122704, A126443, A352859.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] 2^k a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; A[] = 0; Do[A[x] = 1 + (Exp[x] - 1) A[2 x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, 2^j*binomial(i, j)*v[j+1])); v; \\ Seiichi Manyama, Jun 18 2022

E.g.f. A(x) satisfies: A(x) = 1 + (exp(x) - 1) * A(2*x).

a(n) ~ c * n! * 2^(n*(n-1)/2), where c = 1.572986203588985421674040830458773854660492965929302012... - Vaclav Kotesovec, Apr 07 2022

A123227 Expansion of e.g.f.: 2exp(2x) / (3 - exp(2*x)).

Original entry on oeis.org

1, 3, 12, 66, 480, 4368, 47712, 608016, 8855040, 145083648, 2641216512, 52891055616, 1155444326400, 27344999497728, 696933753434112, 19031293222127616, 554336947975618560, 17155693983744196608, 562168282464340672512, 19444889661250162262016

Offset: 0

Philippe Deléham, Oct 06 2006

G. C. Greubel, Table of n, a(n) for n = 0..200
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Eric Weisstein's MathWorld, Polylogarithm.
Eric Weisstein's MathWorld, Lerch Transcendent.

Cf. A000629, A201339, A122704, A009362, A123125, A028246.

a := n -> 2^(n+1)*polylog(-n, 1/3):
seq(round(evalf(a(n),32)), n=0..19); # Peter Luschny, Nov 03 2015
seq(expand(2^(n+1)*polylog(-n,1/3)), n=0..100); # Robert Israel, Nov 03 2015

CoefficientList[Series[2*Exp[2*x]/(3-Exp[2*x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 24 2013 *)
Round@Table[(-1)^(n+1) (LerchPhi[Sqrt[3], -n, 0] + LerchPhi[-Sqrt[3], -n, 0]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

{a(n)=n!*polcoeff(2*exp(2*x+x*O(x^n))/(3 - exp(2*x+x*O(x^n))), n)} /* Paul D. Hanna */

{a(n)=polcoeff(sum(m=0, n, 3^m*m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)} /* Paul D. Hanna */

{Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, (-2)^(n-k)*3^k*Stirling2(n, k)*k!)} /* Paul D. Hanna */

my(x='x+O('x^20)); Vec(serlaplace(2*exp(2*x)/(3-exp(2*x)))) \\ Joerg Arndt, May 06 2013

@CachedFunction
def BB(n, k, x):  # Modified Cardinal B-splines
    if n == 1: return 0 if (x < 0) or (x >= k) else 1
    return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
def EulerianPolynomial(n, k, x):
    if n == 0: return 1
    return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
def A123227(n) : return 3^n*EulerianPolynomial(n, 1, 1/3)
[A123227(n) for n in (0..18)]  # Peter Luschny, May 04 2013

a(n) = abs(A009362(n+1)).
a(n-1) = Sum_{k=1..n} 2^(n-k)*A028246(n,k), n>=1.
a(n) = Sum_{k=0..n} 3^k*A123125(n,k).
From Paul D. Hanna, Nov 30 2011: (Start)
a(n) = 3*A122704(n) for n>0.
a(n) = Sum_{k=0..n} (-2)^(n-k) * 3^k * Stirling2(n,k) * k!.
O.g.f.: Sum_{n>=0} 3^n * n!*x^n / Product_{k=0..n} (1+2*k*x).
O.g.f.: 1/(1 - 3*x/(1-x/(1 - 6*x/(1-2*x/(1 - 9*x/(1-3*x/(1 - 12*x/(1-4*x/(1 - 15*x/(1-5*x/(1 - ...))))))))))), a continued fraction.
(End)
a(n) ~ n! * (2/log(3))^(n+1). - Vaclav Kotesovec, Jun 24 2013
a(n) = 2^n*log(3)*Integral_{x = 0..oo} (ceiling(x))^n * 3^(-x) dx. - Peter Bala, Feb 06 2015
a(n) = (-1)^(n+1)*(LerchPhi(sqrt(3), -n, 0) + LerchPhi(-sqrt(3), -n, 0)) = (-1)^(n+1)*(Li_{-n}(sqrt(3)) + Li_{-n}(-sqrt(3))) - 2*0^n, where Li_n(x) is the polylogarithm. - Vladimir Reshetnikov, Oct 31 2015
a(n) = 2^(n+1)*Li_{-n}(1/3). - Peter Luschny, Nov 03 2015
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 05 2020

Name changed and a(8) corrected by Paul D. Hanna, Nov 30 2011

A326323 A(n, k) = A_{n}(k) where A_{n}(x) are the Eulerian polynomials, square array read by ascending antidiagonals, for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 75, 120, 1, 1, 1, 6, 33, 160, 541, 720, 1, 1, 1, 7, 46, 285, 1456, 4683, 5040, 1, 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1, 1, 1, 9, 78, 679, 5656, 40005, 202672, 545835, 362880, 1

Offset: 0

Peter Luschny, Jun 27 2019

			Array starts:
  k=0: 1, 1, 1,  1,    1,     1,      1,        1,         1, ... [A000012]
  k=1: 1, 1, 2,  6,   24,   120,    720,     5040,     40320, ... [A000142]
  k=2: 1, 1, 3, 13,   75,   541,   4683,    47293,    545835, ... [A000670]
  k=3: 1, 1, 4, 22,  160,  1456,  15904,   202672,   2951680, ... [A122704]
  k=4: 1, 1, 5, 33,  285,  3081,  40005,   606033,  10491885, ... [A255927]
  k=5: 1, 1, 6, 46,  456,  5656,  84336,  1467376,  29175936, ... [A326324]
  k=6: 1, 1, 7, 61,  679,  9445, 158095,  3088765,  68958295, ... [A384525]
  k=7: 1, 1, 8, 78,  960, 14736, 272448,  5881968, 145105920, ... [A384514]
  k=8: 1, 1, 9, 97, 1305, 21841, 440649, 10386817, 279768825, ...
Seen as a triangle:
  [0], 1
  [1], 1, 1
  [2], 1, 1, 1
  [3], 1, 1, 2,  1
  [4], 1, 1, 3,  6,   1
  [5], 1, 1, 4, 13,  24,    1
  [6], 1, 1, 5, 22,  75,  120,     1
  [7], 1, 1, 6, 33, 160,  541,   720,     1
  [8], 1, 1, 7, 46, 285, 1456,  4683,  5040,     1
  [9], 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1

OEIS Wiki, Eulerian polynomials.

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A384525, A384514.

A := (n, k) -> add(combinat:-eulerian1(k, j)*n^j, j=0..k):
seq(seq(A(n-k, k), k=0..n), n=0..10);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n  - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(k!*coeff(ser(n), x, k), k=0..9) od;

a[n_, 0] := 1; a[n_, 1] := n!;
a[n_, k_] := (k - 1)^(n + 1)/k HurwitzLerchPhi[1/k, -n, 0];
(* Alternative: *) a[n_, k_] := Sum[StirlingS2[n, j] (k - 1)^(n - j) j!, {j, 0, n}];
Table[Print[Table[a[n, k], {n, 0, 10}]], {k, 0, 8}]

A(n, k) = Sum_{j=0..k} a(k, j)*n^j where a(k, j) are the Eulerian numbers.
E.g.f.: (n - 1)/(n - exp((n-1)*x)) for n = 0 and n >= 2, 1/(1 - x) if n = 1.
A(n, 0) = 1; A(n, 1) = n!.
A(n, k) = (k - 1)^(n + 1)/k HurwitzLerchPhi(1/k, -n, 0) for k >= 2.
A(n, k) = Sum_{j=0..n} j! * Stirling2(n, j) * (k - 1)^(n - j) for k >= 2.

cp's OEIS Frontend

A123125 Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.

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A131689 Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

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A336950 E.g.f.: 1 / (1 - x * exp(2*x)).

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A255927 a(n) = (3/4) * Sum_{k>=0} (3*k)^n/4^k.

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A326324 a(n) = A_{5}(n) where A_{m}(x) are the Eulerian polynomials as defined in A326323.

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A352860 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k) * 2^k * a(k).

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A123227 Expansion of e.g.f.: 2*exp(2*x) / (3 - exp(2*x)).

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A131689 Triangle of numbers T(n,k) = k!Stirling2(n,k) = A000142(k)A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

A123227 Expansion of e.g.f.: 2exp(2x) / (3 - exp(2*x)).

A355110 Expansion of e.g.f. 2 / (3 - 2x - exp(2x)).

A351756 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 2x)) / (1 - 2x)^2.