A004123 -id:A004123 - cp's OEIS frontend

A004122 Generalized weak orders on n points.

2, 13, 123, 1546, 24283, 457699, 10064848, 252945467, 7151532895, 224661610888, 7763387794649, 292659248485051, 11951855446598278, 525645673381008537, 24769319755329986599, 1244984053628241578058, 66487872534167725541751

Offset: 1

N. J. A. Sloane

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. G. Wagner, Enumeration of generalized weak orders. Arch. Math. (Basel) 39 (1982), no. 2, 147-152.

T. D. Noe, Table of n, a(n) for n = 1..100
C. G. Wagner, Enumeration of generalized weak orders, Preprint, 1980. [Annotated scanned copy]
C. G. Wagner and N. J. A. Sloane, Correspondence, 1980

Cf. A004121, A004123, A000670.

With[{nn=20},Rest[CoefficientList[Series[1/(2-Exp[x]Exp[Exp[x]-1]),{x,0,nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Nov 05 2011 *)

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

Formula and more terms from Vladeta Jovovic, Mar 27 2001

A094503 Triangle read by rows: coefficients d(n,k) of Andre polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 11, 4, 1, 26, 34, 1, 57, 180, 34, 1, 120, 768, 496, 1, 247, 2904, 4288, 496, 1, 502, 10194, 28768, 11056, 1, 1013, 34096, 166042, 141584, 11056, 1, 2036, 110392, 868744, 1372088, 349504, 1, 4083, 349500, 4247720, 11204160, 6213288

Offset: 1

Philippe Deléham, Jun 09 2004

a(n,k) is the number of increasing 0-1-2 trees on [n] with k leaves. An increasing 0-1-2 tree on [n] is an unordered tree on [n], rooted at 1, in which each vertex has <= 2 children and the labels increase along each path from the root. Example: a(4,2)=4 counts the trees with edges as follows, {1->2->3,1->4}, {1->2->4,1->3}, {1->2->4,2->3}, {1->3->4,1->2}. - David Callan, Oct 24 2004

			Triangle begins:
  1
  1
  1  1
  1  4
  1 11   4
  1 26  34
  1 57 180 34
  ...
From _Peter Bala_, Jun 26 2012: (Start)
Recurrence equation: T(6,3) = 3*T(5,3) + 2*T(5,2) = 3*4 + 2*11 = 34.
n = 4: the 5 weighted non-plane increasing 0-1-2 trees on 4 vertices are
.........................................................
..4......................................................
..|......................................................
..3............4............4.............3.......3...4..
..|.........../............/............./.........\./...
..2......2...3........3...2.........4...2........(t)2....
..|.......\./..........\./...........\./............|....
..1.....(t)1.........(t)1..........(t)1.............1....
.........................................................
Hence row polynomial R(4,t) = (1 + 4*t)*t.
(End)

Alois P. Heinz, Rows n = 1..200, flattened
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
F. Bergeron, Ph. Flajolet, and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
David Callan, A note on downup permutations and increasing 0-1-2 trees
Chak-On Chow and Wai Chee Shiu, Counting Simsun Permutations by Descents, Ann. Comb. 15, 625-635 (2011). See p. 627 and comments section in A113897.
Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 35.
Filippo Disanto and Thomas Wiehe, Exact enumeration of cherries and pitchforks in ranked trees under the coalescent model, arXiv preprint arXiv:1112.1295 [math.CO], 2011-2012, see Y(x,z).
Filippo Disanto and Thomas Wiehe, Exact enumeration of cherries and pitchforks in ranked trees under the coalescent model, Math. Biosci. 242 (2013), no. 2, 195-200.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
Dominique Foata and Guoniu Han, Arbres minimax et polynomes d'André .
D. Foata and Guo-Niu Han, Arbres minimax et polynomes d'André, Adv. in Appl. Math., 27 (2001), no.2-3, 367-389.
Guo-Niu Han and Shi-Mei Ma, Derivatives, Eulerian polynomials and the g-indexes of Young tableaux, arXiv:2006.14064 [math.CO], 2020.
Shi-Mei Ma, Enumeration of permutations by number of alternating runs, Discrete Math., 313 (2013), 1816-1822.
Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021
Shi-Mei Ma and Yeong-Nan Yeh, The Peak Statistics on Simsun Permutations, Elect. J. Combin., 23 (2016), P2.14; arXiv preprint, arXiv:1601.06505 [math.CO], 2016.
E. Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013.

Cf. A000012, A000295, A008292, A101280, A113897.

A094503:=proc(n,k) options remember: if(n=1 and k=1) then RETURN(1) elif(1<=k and k<=floor((n+1)/2) and n>=1) then RETURN(k*A094503(n-1,k)+(n+2-2*k)*A094503(n-1,k-1)) else RETURN(0) fi: end; seq(seq(A094503(n,k),k=1..floor((n+1)/2)),n=1..14);

t[1, 1] = 1; t[n_, k_] /; Not[1 <= k <= (n+1)/2] = 0; t[n_, k_] := t[n, k] = k*t[n-1, k] + (n+2-2*k)*t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 1, (n + 1)/2}] // Flatten (* Jean-François Alcover, Nov 22 2012, after Maple *)

def p(n) :
    s = var('s'); u = sqrt(s^2-2)
    egf = u*x-2*ln((exp(u*x)*(1-s/u)+s/u+1)/2)
    return factorial(n+2)*egf.series(x,n+4).coefficient(x,n+2)
def A094503_row(n) : return [p(n).coefficient(s,n-2*i) for i in (0..n//2)]
for n in (0..6): print(A094503_row(n)) # Peter Luschny, Jul 01 2012

D(1, n) = A000111(n), Euler or up/down numbers. D(1/2, n) = A000142(n)*(1/2)^n. D(1/4, n) = A080795(n)*(1/4)^n.
From Peter Bala, Jun 26 2012: (Start):
Recurrence equation: T(n,k) = k*T(n-1,k) + (n+2-2*k)*T(n-1,k-1) for n >= 1 and 1 <= k <= floor((n+1)/2).
Let r(t) = sqrt(1-2*t) and w(t) = (1-r(t))/(1+r(t)). The e.g.f. is F(t,z) = r(t)*(1 + w(t)*exp(r(t)*z))/(1 - w(t)*exp(r(t)*z)) = 1 + t*z + t*z^2/2! + (t+t^2)*z^3/3! + (t+4*t^2)*z^4/4! + ... (Foata and Han, 2001, section 7).
Note that (F(2*t,z) - 1)/(2*t) is the e.g.f. for A101280.
The modified e.g.f. A(t,z) := (F(t,z) - 1)/t = z + z^2/2! + (1+t)*z^3/3! + (1+4*t)*z^4/4! + ... satisfies the autonomous partial differential equation dA/dz = 1 + A + 1/2*t*A^2 with A(t,0) = 1. It follows that the inverse function A(t,z)^(-1) may be expressed as an integral: A(t,z)^(-1) = Integral_{x = 0..z} 1/(1+x+1/2*t*x^2) dx.
Applying [Dominici, Theorem 4.1] to invert the integral gives the following method for calculating the row polynomials R(n,t) of the table: let f(t,x) = 1+x+1/2*t*x^2 and let D be the operator f(t,x)*d/dx. Then R(n+1,t) = t*D^n(f(t,x)) evaluated at x = 0.
By Bergeron et al., Theorem 1, the shifted row polynomial 1/t*R(n,t) is the generating function for rooted non-plane increasing 0-1-2 trees on n vertices, where the vertices of outdegree 2 have weight t and all other vertices have weight 1. An example is given below.
1/(2*t)*(1+t)^(n+1)*R(n,2*t/(1+t)^2) = the n-th Eulerian polynomial of A008292. For example, n = 5 gives 1/(2*t)*(1+t)^6*R(5,2*t/(1+t)^2) = 1 + 26*t + 66*t^2 + 26*t^3 + t^4.
A000142(n) = 2^n*R(n,1/2); A080795(n) = 4^n*R(n,1/4);
A000670(n) = 3/4*3^n*R(n,4/9); A004123(n+1) = 5/6*5^n*R(n,12/25).
(End)
There is a second family of polynomials which also matches the data and is different from the André polynomials as defined by Foata and Han (2001), formula 3.5. Let u = sqrt(s^2-2) and F(s,x) = u*x-2*log((exp(u*x)*(1-s/u)+s/u+1)/2), then for n>=0 the sequence of polynomials p_{n}(s) = (n+2)!*[x^(n+2)]F(s,x) starts 1, s, s^2+1, s^3+4*s, s^4+11*s^2+4, s^5+26*s^3+34*s, s^6+57*s^4+180*s^2+34, ... and the nonzero coefficients of these polynomials in descending order coincide with the sequence a(n). p_{n}(0) is an aerated version of the reduced tangent numbers, p_{2*n}(0) = A002105(n+1) for n>=0. In contrast, the André polynomials vanish at t=0 except for n=0. - Peter Luschny, Jul 01 2012
T(n,k) = A008303(n,k)/2^(n-k). - Ammar Khatab, Aug 17 2024

A154694 Triangle read by rows: T(n,k) = ((3/2)^k2^n + (2/3)^k3^n)*A008292(n+1,k+1).

Original entry on oeis.org

2, 5, 5, 13, 48, 13, 35, 330, 330, 35, 97, 2028, 4752, 2028, 97, 275, 11970, 54360, 54360, 11970, 275, 793, 69840, 557388, 1043712, 557388, 69840, 793, 2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315, 6817, 2388516, 51011136, 247761072, 404844480, 247761072, 51011136, 2388516, 6817

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

			Triangle begins as:
     2;
     5,      5;
    13,     48,      13;
    35,    330,     330,       35;
    97,   2028,    4752,     2028,       97;
   275,  11970,   54360,    54360,    11970,     275;
   793,  69840,  557388,  1043712,   557388,   69840,    793;
  2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A 34 (3) (1986) 1986, page 2502, (FIG. 3)

Cf. A004123 (row sums), A154693, A256890.

A154694:= func< n,k | (2^(n-k)*3^k+2^k*3^(n-k))*EulerianNumber(n+1, k) >;
[A154694(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154694 := proc(n,m)
    (3^m*2^(n-m)+2^m*3^(n-m))*A008292(n+1,m+1) ;
end proc:
seq(seq( A154694(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Mar 11 2024

T[n_, k_, p_, q_] := (p^(n - k)*q^k + p^k*q^(n - k))*Eulerian[n+1,k];
Table[T[n,k,2,3], {n,0,12}, {k,0,n}]//Flatten

from sage.all import *
from sage.combinat.combinat import eulerian_number
def A154694(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*eulerian_number(n+1,k)
print(flatten([[A154694(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Sum_{k=0..n} T(n, k) = A004123(n+2).

Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010

A320352 Expansion of e.g.f. (exp(x) - 1)/(exp(x) - exp(2*x) + 1).

Original entry on oeis.org

0, 1, 3, 19, 159, 1651, 20583, 299419, 4977759, 93097891, 1934655063, 44224195819, 1102820674959, 29792843865331, 866769668577543, 27018340680076219, 898343366411181759, 31736205208791131971, 1187110673532381604023, 46871464129796857140619, 1948059531745350527058159

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

nonn

From Peter Bala, Aug 19 2025: (Start)

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [0, 1, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, ...] with an apparent period of 6 = phi(9) beginning at n = 2. Cf. A004123.(End)

Seiichi Manyama, Table of n, a(n) for n = 0..401

Cf. A000045, A000556, A000557, A005443, A005445, A048993, A263576.

seq(n!*coeff(series((exp(x) - 1)/(exp(x) - exp(2*x) + 1), x=0, 22), x, n), n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 20; CoefficientList[Series[(Exp[x] - 1)/(Exp[x] - Exp[2 x] + 1), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] Fibonacci[k] k!, {k, 0, n}], {n, 0, 20}]

E.g.f.: (1 + sinh(x) - cosh(x))/(1 - 2*sinh(x)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*Fibonacci(k)*k!.
a(n) ~ n! / (sqrt(5) * phi^2 * (log(phi))^(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 12 2018

A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 10, 3, 1, 0, 75, 74, 21, 4, 1, 0, 541, 730, 219, 36, 5, 1, 0, 4683, 9002, 3045, 484, 55, 6, 1, 0, 47293, 133210, 52923, 8676, 905, 78, 7, 1, 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1, 0, 7087261, 45375130, 26857659, 5227236, 544505, 39390, 2359, 136, 9, 1

Offset: 0

Peter Luschny, May 21 2021

The array rows are recursively generated by applying the Akiyama-Tanigawa algorithm to the powers (see the Python implementation below). In this way the array becomes the image of A004248 under the AT-transformation when applied to the columns of A004248. This makes the array closely linked to A371761, which is generated in the same way, but applied to the rows of A004248. - Peter Luschny, Apr 27 2024

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,      1;
[3] 0, 3,      2,       1;
[4] 0, 13,     10,      3,       1;
[5] 0, 75,     74,      21,      4,      1;
[6] 0, 541,    730,     219,     36,     5,     1;
[7] 0, 4683,   9002,    3045,    484,    55,    6,    1;
[8] 0, 47293,  133210,  52923,   8676,   905,   78,   7,   1;
[9] 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1;
.
Seen as an array A(n, k) = T(n + k, n):
[0] [1, 0,   0,    0,     0,       0,         0, ...  A000007
[1] [1, 1,   3,   13,    75,     541,      4683, ...  A000670
[2] [1, 2,  10,   74,   730,    9002,    133210, ...  A004123
[3] [1, 3,  21,  219,  3045,   52923,   1103781, ...  A032033
[4] [1, 4,  36,  484,  8676,  194404,   5227236, ...  A094417
[5] [1, 5,  55,  905, 19855,  544505,  17919055, ...  A094418
[6] [1, 6,  78, 1518, 39390, 1277646,  49729758, ...  A094419
[7] [1, 7, 105, 2359, 70665, 2646007, 118893705, ...  A238464

Variant of the array is A094416 (which has column 0 and row 0 missing).
The coefficients of the Fubini polynomials are A131689.
Cf. A094420 (main diagonal of array), A372346 (row sums), A004248, A371761.

F := proc(n) option remember; if n = 0 then return 1 fi:
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
seq(seq(subs(x = k, F(n - k)), k = 0..n), n = 0..10);

F[n_] := F[n] = If[n == 0, 1,
   Expand[Sum[Binomial[n, k]*F[n - k]*x, {k, 1, n}]]];
Table[Table[F[n - k] /. x -> k, {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 06 2024, after Peter Luschny *)

# Computes the triangle.
@cached_function
def F(n):
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
def Fval(n): return [F(n - k).substitute(x = k) for k in (0..n)]
for n in range(10): print(Fval(n))

# Computes the square array using the Akiyama-Tanigawa algorithm.
def ATFubini(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = (n + 1)**k  # Chancing this to R[k] = k**n generates A371761.
        for j in range(k, 0, -1):
            R[j - 1] = j * (R[j] - R[j - 1])
        A[k] = R[0]
    return A
for n in range(8): print([n], ATFubini(n, 7))  # Peter Luschny, Apr 27 2024

T(n, k) = (n - k)! * [x^(n - k)] (1 / (1 + k * (1 - exp(x)))).

T(2*n, n) = A094420(n).

A004121 Generalized weak orders on n points.

Original entry on oeis.org

2, 16, 208, 3968, 109568, 4793344, 410662912, 82657083392, 38274970222592, 37590755515826176, 75458309991776124928, 305873605165090925969408, 2491832958314452159507202048, 40704585435508852018947014262784

Offset: 1

N. J. A. Sloane

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. G. Wagner, Enumeration of generalized weak orders. Arch. Math. (Basel) 39 (1982), no. 2, 147-152.

Vincenzo Librandi, Table of n, a(n) for n = 1..80
C. G. Wagner, Enumeration of generalized weak orders, Preprint, 1980. [Annotated scanned copy]
C. G. Wagner and N. J. A. Sloane, Correspondence, 1980

Cf. A004122, A004123, A000670 (asymmetric generalized weak orders on n points).

max = 14; f[x_] := 1/(1 - Sum[(2^(i*(i+1)/2)*x^i)/i!, {i, 1, max}]); Drop[ CoefficientList[ Series[f[x], {x, 0, max}], x]*Range[0, max]!, 1] (* Jean-François Alcover, Oct 21 2011, after g.f. *)

E.g.f.: 1/(1 - Sum_{i >= 1} 2^binomial(i+1, 2)*x^i/i!).

Formula and more terms from Vladeta Jovovic, Mar 27 2001

A195205 Triangle of coefficients of a sequence of binomial type polynomials.

Original entry on oeis.org

3, 6, 9, 30, 54, 27, 222, 468, 324, 81, 2190, 5130, 4320, 1620, 243, 27006, 68400, 65610, 30780, 7290, 729, 399630, 1076166, 1135890, 618030, 187110, 30618, 2187, 6899262, 19532268, 22212792, 13471920, 4796820, 1020600, 122472, 6561

Offset: 1

Peter Bala, Sep 13 2011

Define a polynomial sequence P_n(x) by means of the recursion
P_(n+1)(x) = x*(P_n(x) + 2*P_n(x+1)), with P_0(x) = 1.
The first few values are
P_1(x) = 3*x, P_2(x) = 3*x*(3*x + 2),
P_3(x) = 3*x*(9*x^2 + 18*x + 10),
P_4(x) = 3*x*(27*x^3 + 108*x^2 + 156*x + 74).
The present table shows the coefficients of these polynomials (excluding P_0(x)) in ascending powers of x. Compare with A195204.
Triangle T(n,k) (1 <= k <= n), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, 3, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011

			Triangle begins
n\k|.....1.......2......3......4......5......6
==============================================
..1|.....3
..2|.....6.......9
..3|....30......54.....27
..4|...222.....468....324.....81
..5|..2190....5130...4320...1620....243
..6|.27006...68400..65610..30780...7290....729
...
Triangle (0, 2, 3, 4, 6, 6, 9, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, ...) begins:
1;
0,     3;
0,     6,     9;
0,    30,    54,    27;
0,   222,   468,   324,    81;
0,  2190,  5130,  4320,  1620,   243;
0, 27006, 68400, 65610, 30780,  7290,   729;
... - _Philippe Deléham_, Dec 22 2011

Wikipedia, Binomial type

Cf. A004123 ((1/3)*column 1), A050351 ((1/3)*row sums), A179929 (delta operator coeffs.), A195204.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0,3,polylog(-n, 2/3)), 10); # Peter Luschny, Jan 29 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[If[# == 0, 3, PolyLog[-#, 2/3]]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

E.g.f.: F(x,z) := (exp(z)/(3 - 2*exp(z)))^x = 1 + 3*x*z + (6*x + 9*x^2)*z^2/2! + (30*x + 54*x^2 + 27*x^3)*z^3/3! + ....
The generating function F(x,z) = Sum_{n>=0} P_n(x)*z^n/n! satisfies the partial differential equation d/dz(F(x,z)) = x*F(x,z) + 2*x*F(x+1,z). Hence the row generating polynomials P_n(x) satisfy the recurrence P_(n+1)(x) = x*(P_n(x) + 2*P_n(x+1)), with P_0(x) = 1. The form of the e.g.f. shows that the polynomials P_n(x) are a sequence of binomial type. In what follows we denote P_n(x) by x^[n].
Relation with rising factorials
x^[n] = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n,k)*3^k*x*(x+1)*...*(x+k-1),
and its inverse formula
3^n*x*(x+1)*...*(x+n-1) = Sum_{k=1..n} |Stirling1(n,k)|*x^[k].
The delta operator D*:
The row polynomials form a polynomial sequence of binomial type. If D denotes the derivative operator 1/3*d/dx then the associated delta operator D* is given by D* = D - 2*D^2/2! + 2*D^3/3! + 6*D^4/4! - 30*D^5/5! - ..., where the sequence of coefficients [1, -2, 2, 6, -30, -42, 882, ...] equals (-1)^n*A179929(n). D* is the lowering operator for the row polynomials, that is, (D*)x^[n] = n*x^[n-1].
Generalized Dobinski formula:
exp(-x)*Sum_{k >= 1} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,3*x),
where the Bell (or exponential) polynomials are defined as
Bell(n,x) := Sum_{k = 1..n} Stirling2(n,k)*x^k.
Relation with the Bell polynomials:
The alternating n-th row entries (-1)^(n+k)*T(n,k) are the connection coefficients expressing the polynomial Bell(n,3*x) as a linear combination of Bell(k,x), 1 <= k <= n. For example for row 4:
Bell(4,3*x) = -222*Bell(1,x) + 468*Bell(2,x) - 324*Bell(3,x) + 81*Bell(4,x).
Generalized Bernoulli summation formula:
We have the following generalization of Bernoulli's formula for the sum of the powers of integers:
3*Sum_{k = 1..n} k^[p] = 1/(p+1)*Sum_{k = 0..p} (-1)^k * binomial(p+1,k)*B_k*n^[p+1-k], where B_k =[1, -1/2, 1/6, 0, -1/30, ...] denotes the sequence of Bernoulli numbers.
Relation with other sequences:
Row sums = 3*A050351(n) for n >= 1. Column 1 = 3*A004123.
T(n,k) = A185285(n,k)*3^k. - Philippe Deléham, Feb 17 2013
Also the Bell transform of 3*A004123. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016
Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 3*x*z/(1 - 2*z/(1 - 3*(x + 1)*z/(1 - 4*z/(1 - 3*(x + 2)*z/(1 - 6*z/(1 - 3*(x + 3)*z/(1 - 8*z/(1 - ... ))))))))). - Peter Bala, Dec 12 2024

A238465 Generalized ordered Bell numbers Bo(8,n).

Original entry on oeis.org

1, 8, 136, 3464, 117640, 4993928, 254396296, 15119104904, 1026912225160, 78468091562888, 6662087721342856, 622186077361470344, 63389713864392140680, 6996476832548305415048, 831619554631233264449416, 105909083171031626820475784

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 8 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(9 - 8*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t = 30; Range[0, t]! CoefficientList[Series[1/(9 - 8 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(9 - 8*exp(x)).
a(n) ~ n! / (9*(log(9/8))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 8*a(n-1) - 9*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A384324 Expansion of Product_{k>=1} 1/(1 - k*x)^((2/3)^k).

Original entry on oeis.org

1, 6, 33, 200, 1428, 12408, 132604, 1730160, 27043866, 495026316, 10388326986, 245555445888, 6446710871724, 185904786328920, 5836500883321164, 198054400887909264, 7220679972923312487, 281402128806812402490, 11671796413017231008663

Offset: 0

Seiichi Manyama, May 26 2025

nonn

Cf. A084785, A384305, A384325, A384326.

Cf. A004123, A090351, A090352.

terms = 20; A[] = 1; Do[A[x] = -2*A[x] + 3*A[x/(1-x)]^(2/3) / (1-x)^2 + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 27 2025 *)

my(N=20, x='x+O('x^N)); Vec(exp(3*sum(k=1, N, sum(j=0, k, 2^j*j!*stirling(k, j, 2))*x^k/k)))

G.f. A(x) satisfies A(x) = A(x/(1-x))^(2/3) / (1-x)^2.
G.f.: exp(3 * Sum_{k>=1} A004123(k+1) * x^k/k).
G.f.: B(x)^6, where B(x) is the g.f. of A090351.
a(n) ~ (n-1)! / log(3/2)^(n+1). - Vaclav Kotesovec, May 27 2025

A384362 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 2^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 10, 4, 1, 1, 74, 148, 8, 1, 1, 730, 13540, 2440, 16, 1, 1, 9002, 2308756, 3087368, 42256, 32, 1, 1, 133210, 632363044, 10208479240, 778026256, 752800, 64, 1, 1, 2299754, 253970683348, 69754997963528, 52520969994256, 207633589664, 13660480, 128, 1

Offset: 0

Seiichi Manyama, May 27 2025

			Square array begins:
  1,  1,     1,         1,              1, ...
  1,  2,    10,        74,            730, ...
  1,  4,   148,     13540,        2308756, ...
  1,  8,  2440,   3087368,    10208479240, ...
  1, 16, 42256, 778026256, 52520969994256, ...

Columns k=0..2 give A000012, A000079, A098270.
Rows n=0..1 give A000012, A004123(k+1).
Cf. A262809, A384364.

a(n, k) = sum(i=0, k*n, 2^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));

A(n,k) = (1/3) * Sum_{j>=0} (2/3)^j * binomial(j,n)^k.

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A004122 Generalized weak orders on n points.

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A094503 Triangle read by rows: coefficients d(n,k) of Andre polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.

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A154694 Triangle read by rows: T(n,k) = ((3/2)^k*2^n + (2/3)^k*3^n)*A008292(n+1,k+1).

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A320352 Expansion of e.g.f. (exp(x) - 1)/(exp(x) - exp(2*x) + 1).

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A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

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A004121 Generalized weak orders on n points.

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A195205 Triangle of coefficients of a sequence of binomial type polynomials.

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A154694 Triangle read by rows: T(n,k) = ((3/2)^k2^n + (2/3)^k3^n)*A008292(n+1,k+1).

A384362 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 2^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.