A004123 -id:A004123 - cp's OEIS frontend

A088222 Coefficient of x^n in g.f.^n is A004123(n).

1, 2, 3, 10, 69, 678, 8496, 128316, 2258262, 45292494, 1018882779, 25399668480, 694999352141, 20710476430548, 667708554093132, 23159551588872624, 860001996926543616, 34043670528120810846, 1431191816223150995395

Offset: 0

Michael Somos, Sep 24 2003

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..320

{a(n)=polcoeff(x/serreverse(x*exp(sum(m=1, n+1, sum(k=0, m, stirling(m, k, 2)*(2^k)*k!)*x^m/m +x^2*O(x^n)))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Feb 11 2015

a(n) ~ 4 * (n-1)! / (27 * (log(3/2))^(n+1)). - Vaclav Kotesovec, Feb 11 2015, updated Feb 18 2017

A167141 G.f.: Sum_{n>=0} A004123(n)^2log(1+x)^n/n! where 1/(1-2x) = Sum_{n>=0} A004123(n)log(1+x)^n/n!.

Original entry on oeis.org

1, 4, 48, 864, 20880, 632448, 23018688, 978179328, 47529084096, 2598928566336, 157937795847936, 10559489876375040, 770269715428025088, 60876094422772800000, 5181654464327251948032, 472584847824904789910016

Offset: 0

Paul D. Hanna, Nov 04 2009

nonn

CONJECTURE: For all integer m>0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

In this case, m=2 and L(n) = A004123(n), which is the number of generalized weak orders on n points.

			G.f.: A(x) = 1 + 4*x + 48*x^2 + 864*x^3 + 20880*x^4 + 632448*x^5 +...
Illustrate A(x) = Sum_{n>=0} A004123(n)^2 * log(1+x)^n/n!:
A(x) = 1 + 2^2*log(1+x) + 10^2*log(1+x)^2/2! + 74^2*log(1+x)^3/3! + 730^2*log(1+x)^4/4! + 9002^2*log(1+x)^5/5! +...+ A004123(n)^2*log(1+x)^n/n! +...
where the e.g.f. of A004123 is 1/(3 - 2*exp(x)) and thus:
1/(1-2x) = 1 + 2*log(1+x) + 10*log(1+x)^2/2! + 74*log(1+x)^3/3! + 730*log(1+x)^4/4! + 9002*log(1+x)^5/5! +...+ A004123(n)*log(1+x)^n/n! +...

Cf. A004123, variants: A167139, A167138, A101370.

{A004123(n)=sum(k=0,n,2^k*stirling(n, k, 2)*k!)}
{a(n)=polcoeff(sum(m=0,n,A004123(m)^2*log(1+x+x*O(x^n))^m/m!),n)}

A174278 Partial sums of A004123.

Original entry on oeis.org

1, 3, 13, 87, 817, 9819, 143029, 2442783, 47817913, 1054997475, 25895101885, 699790692519, 20644163034049, 660099532324971, 22739373410768581, 839552217608213295, 33071685749731393225, 1384473468760664408307

Offset: 1

Jonathan Vos Post, Mar 15 2010

Partial sums of the number of generalized weak orders on n points. Equivalently, partial sums of the number of bipartitional relations on a set of cardinality n.

G. C. Greubel, Table of n, a(n) for n = 1..390

Cf. A004123.

A004123[n_]:= A004123[n]= Sum[2^k*k!*StirlingS2[n-1,k], {k,0,n-1}];
A174278[n_]:= Sum[A004123[j], {j,0,n}];
Table[A174278[n], {n,30}] (* G. C. Greubel, Mar 25 2022 *)

def A004123(n): return sum(stirling_number2(n-1, k)*(2^k)*factorial(k) for k in (0..n-1))
def A174278(n): return sum(A004123(j) for j in (0..n))
[A174278(n) for n in (1..30)] # G. C. Greubel, Mar 25 2022

a(n) = Sum_{i=1..n} A004123(i).
a(n) = Sum_{i=1..n} Sum_{k >= 0} (k^n*(2/3)^k)/3.
a(n) = Sum_{i=1..n} Sum_{k = 0..n} Stirling2(n,k)*(2^k)*k!.

A088794 Coefficient of x^n in A(x)^(2n) is A004123(n); self-convolution is A088222.

Original entry on oeis.org

1, 1, 1, 4, 30, 305, 3905, 59828, 1063728, 21497921, 486476766, 12184618776, 334684804952, 10005219881472, 323438539163521, 11244331792094312, 418375698771595037, 16590419690069321454, 698526596162530976512

Offset: 0

Paul D. Hanna, Oct 23 2003

nonn

Cf. A088222, A004123.

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

A050351 Number of 3-level labeled linear rooted trees with n leaves.

Original entry on oeis.org

1, 1, 5, 37, 365, 4501, 66605, 1149877, 22687565, 503589781, 12420052205, 336947795317, 9972186170765, 319727684645461, 11039636939221805, 408406422098722357, 16116066766061589965, 675700891505466507541

Offset: 0

Christian G. Bower, Oct 15 1999

nonn

Lists of lists of sets.

			G.f. = 1 + x + 5*x^2 + 37*x^3 + 365*x^4 + 4501*x^5 + 66605*x^6 + ...

T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

G. C. Greubel, Table of n, a(n) for n = 0..390
Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.
S. Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938 [math.CO], 2013.
Marian Muresan, A concrete approach to classical analysis, CMS Books in Mathematics (2009) Table 10.2
Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Index entries for sequences related to rooted trees

Cf. A000670, A050352-A050359.

Equals 1/2 * A004123(n) for n>0.

with(combstruct); SeqSeqSetL := [T, {T=Sequence(S), S=Sequence(U,card >= 1), U=Set(Z,card >=1)},labeled];

With[{nn=20},CoefficientList[Series[(2-E^x)/(3-2*E^x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 29 2012 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1/(2 - 1/(2 - Exp[x])), {x, 0, n}]]; (* Michael Somos, Nov 28 2014 *)

{a(n) = if( n<0, 0, n! * polcoeff( 1/(2 - 1/(2 - exp(x + x * O(x^n)))), n))};

{a(n)=if(n==0, 1, (1/6)*round(suminf(k=1, k^n * (2/3)^k *1.)))} \\ Paul D. Hanna, Nov 28 2014

A050351 = lambda n: sum(stirling_number2(n,k)*(2^(k-1))*factorial(k) for k in (0..n)) if n>0 else 1
[A050351(n) for n in (0..17)] # Peter Luschny, Jan 18 2016

E.g.f.: (2-exp(x))/(3-2*exp(x)).
a(n) is asymptotic to (1/6)*n!/log(3/2)^(n+1). - Benoit Cloitre, Jan 30 2003
For m-level trees (m>1), e.g.f. is (m-1-(m-2)*e^x)/(m-(m-1)*e^x) and number of trees is 1/(m*(m-1))*sum(k>=0, (1-1/m)^k*k^n). Here m=3, so a(n)=(1/6)*sum(k>=0, (2/3)^k*k^n) (for n>0). - Benoit Cloitre, Jan 30 2003
a(n) = Sum_{k=1..n} Stirling2(n, k)*k!*2^(k-1). - Vladeta Jovovic, Sep 28 2003
Recurrence: a(n+1) = 1 + 2*Sum_{j=1..n} binomial(n+1, j)*a(j). - Jon Perry, Apr 25 2005
With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)}(n!/(prod_{j=1}^{p(i)}p(i, j)!))*(p(i)!/(prod_{j=1}^{d(i)} m(i, j)!))*2^(p(i)-1). - Thomas Wieder, May 18 2005
Let f(x) = (1+x)*(1+2*x). Let D be the operator g(x) -> d/dx(f(x)*g(x)). Then for n>=1, a(n) = D^(n-1)(1) evaluated at x = 1/2. Compare with the result A000670(n) = D^(n-1)(1) at x = 0. See also A194649. - Peter Bala, Sep 05 2011
E.g.f.: 1 + x/(G(0)-3*x) where G(k)= x + k + 1 - x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 11 2012
a(n) = (1/6) * Sum_{k>=1} k^n * (2/3)^k for n>0. - Paul D. Hanna, Nov 28 2014
E.g.f. A(x) satisfies 0 = 2 - A'(x) - 7*A(x) + 6*A(x)^2. - Michael Somos, Nov 28 2014

A094417 Generalized ordered Bell numbers Bo(4,n).

Original entry on oeis.org

1, 4, 36, 484, 8676, 194404, 5227236, 163978084, 5878837476, 237109864804, 10625889182436, 523809809059684, 28168941794178276, 1641079211868751204, 102961115527874385636, 6921180217049667005284, 496267460209336700111076, 37807710659221213027893604

Offset: 0

Ralf Stephan, May 02 2004

nonn

Fourth row of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Cf. A000670, A004123, A032033, A050353, A094418, A094419, A131689.

Cf. A346983, A354242, A365567.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(5 - 4*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Bruno Berselli, Mar 17 2014

a:= proc(n) option remember;
      `if`(n=0, 1, 4* add(binomial(n, k) *a(k), k=0..n-1))
    end:
seq(a(n), n=0..20);

max = 16; f[x_] := 1/(5-4*E^x); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Nov 14 2011, after g.f. *)

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(5 - 4*exp(x)))) \\ Joerg Arndt, Jan 15 2024

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094417(k): return A094416(4,k)
[A094417(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(5 - 4*exp(x)).
a(n) = 4 * A050353(n) for n>0.
a(n) = Sum_{k=0..n} A131689(n,k) * 4^k. - Philippe Deléham, Nov 03 2008
E.g.f.: A(x) with A_n = 4 * Sum_{k=0..n-1} C(n,k) * A_k; A_0 = 1. - Vladimir Kruchinin, Jan 27 2011
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 10*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) = log(5/4)*int {x = 0..inf} (floor(x))^n * (5/4)^(-x) dx. - Peter Bala, Feb 14 2015
a(0) = 1; a(n) = 4 * a(n-1) - 5 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/5 * Li_{-n}(5/4), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/5) * Sum_{k>=0} k^n * (4/5)^k.
a(n) = (4/5) * Sum_{k=0..n} 5^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

A094416 Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).

Original entry on oeis.org

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

Offset: 1

Ralf Stephan, May 02 2004

Also, r times the number of (r+1)-level labeled linear rooted trees with n leaves.
"AIJ" (ordered, indistinct, labeled) transform of {r,r,r,...}.
Stirling transform of r^n*n!, i.e. of e.g.f. 1/(1-r*x).
Also, Bo(r,s) is ((x*d/dx)^n)(1/(1+r-r*x)) evaluated at x=1.
r-th ordered Bell polynomial (A019538) evaluated at n.
Bo(r,n) is the n-th moment of a geometric distribution with probability parameter = 1/(r+1).  Here, geometric distribution is the number of failures prior to the first success. - Geoffrey Critzer, Jan 01 2019
Row r (starting at r=0), Bo(r+1, n), is the Akiyama-Tanigawa algorithm applied to the powers of r+1. See Python program below. - Shel Kaphan, May 03 2024

			Array begins as:
  1,  3,   13,    75,     541,     4683,      47293, ...
  2, 10,   74,   730,    9002,   133210,    2299754, ...
  3, 21,  219,  3045,   52923,  1103781,   26857659, ...
  4, 36,  484,  8676,  194404,  5227236,  163978084, ...
  5, 55,  905, 19855,  544505, 17919055,  687978905, ...
  6, 78, 1518, 39390, 1277646, 49729758, 2258233998, ...

G. C. Greubel, Antidiagonals n = 1..50, flattened
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution, arXiv:quant-ph/0303030, 2003.
C. G. Bower, Transforms

Rows 1-10 are A000670, A004123, A032033, A094417, A094418, A094419, A238464, A238465, A238466, A238467.
Columns include A014105, A094421.
Main diagonal is A094420.
Antidiagonal sums are A094422.
Cf. A019538, A131689, A344499.

A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
[A094416(n-k+1,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jan 12 2024

Bo[, 0]=1; Bo[r, n_]:= Bo[r, n]= r*Sum[Binomial[n,k] Bo[r,n-k], {k, n}];
Table[Bo[r-n+1, n], {r, 10}, {n, r}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

# The Akiyama-Tanigawa algorithm applied to the powers of r + 1
# generates the rows. Adds one row (r=0) and one column (n=0).
# Adapted from Peter Luschny on A371568.
def f(n, r): return (r + 1)**n
def ATtransform(r, len, f):
  A = [0] * len
  R = [0] * len
  for n in range(len):
      R[n] = f(n, r)
      for j in range(n, 0, -1):
          R[j - 1] = j * (R[j] - R[j - 1])
      A[n] = R[0]
  return A
for r in range(8): print([r], ATtransform(r, 8, f)) # Shel Kaphan, May 03 2024

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
flatten([[A094416(n-k+1,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(1 + r*(1 - exp(x))).
Bo(r, n) = Sum_{k=0..n} k!*r^k*Stirling2(n, k) = 1/(r+1) * Sum_{k>=1} k^n * (r/(r+1))^k, for r>0, n>0.
Recurrence: Bo(r, n) = r * Sum_{k=1..n} C(n, k)*Bo(r, n-k), with Bo(r, 0) = 1.
Bo(r,0) = 1, Bo(r,n) = r*Bo(r,n-1) - (r+1)*Sum_{j=1..n-1} (-1)^j * binomial(n-1,j) * Bo(r,n-j). - Seiichi Manyama, Nov 17 2023

Offset corrected by Geoffrey Critzer, Jan 01 2019

A305404 Expansion of Sum_{k>=0} (2k - 1)!!x^k/Product_{j=1..k} (1 - j*x).

Original entry on oeis.org

1, 1, 4, 25, 217, 2416, 32839, 527185, 9761602, 204800551, 4801461049, 124402647370, 3529848676237, 108859319101261, 3625569585663484, 129689000146431205, 4958830249864725997, 201834650901695603296, 8712774828941647677019, 397596632650906687905565

Offset: 0

Ilya Gutkovskiy, May 31 2018

Stirling transform of A001147.

Alois P. Heinz, Table of n, a(n) for n = 0..391
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000670, A001147, A004123, A305405, A346982 - A346985.

b:= proc(n, m) option remember;
     `if`(n=0, doublefactorial(2*m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 04 2021

nmax = 19; CoefficientList[Series[Sum[(2 k - 1)!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[1/Sqrt[3 - 2 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] (2 k - 1)!!, {k, 0, n}], {n, 0, 19}]

E.g.f.: 1/sqrt(3 - 2*exp(x)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*(2*k - 1)!!.
a(n) ~ sqrt(2/3) * n^n / ((log(3/2))^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jul 01 2018
Conjecture: a(n) = Sum_{k>=0} k^n * binomial(2*k,k) / (2^k * 3^(k + 1/2)). - Diego Rattaggi, Oct 11 2020
O.g.f. conjectural: 1/(1 - x/(1 - 3*x/(1 - 3*x/(1 - 6*x/(1 - 5*x/(1 - 9*x/(1 - 7*x/(1 - ... - (2*n-1)*x/(1 - 3*n*x/(1 - ... )))))))))) - a continued fraction of Stieltjes-type. - Peter Bala, Dec 06 2020
a(0) = 1; a(n) = Sum_{k=1..n} (2 - k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A201339 Expansion of e.g.f. exp(x) / (3 - 2*exp(x)).

Original entry on oeis.org

1, 3, 15, 111, 1095, 13503, 199815, 3449631, 68062695, 1510769343, 37260156615, 1010843385951, 29916558512295, 959183053936383, 33118910817665415, 1225219266296167071, 48348200298184769895, 2027102674516399522623, 89990106205541777926215, 4216915299772659459872991

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 3*x + 15*x^2/2! + 111*x^3/3! + 1095*x^4/4! + 13503*x^5/5! + ...
O.g.f.: A(x) = 1 + 3*x + 15*x^2 + 111*x^3 + 1095*x^4 + 13503*x^5 + ...
where A(x) = 1 + 3*x/(1+x) + 2!*3^2*x^2/((1+x)*(1+2*x)) + 3!*3^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*3^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000629, A004123, A050351, A123227.

[&+[(-1)^(n-j)*3^j*Factorial(j)*StirlingSecond(n,j): j in [0..n]]: n in [0..20]]; // G. C. Greubel, Jun 08 2020

seq(coeff(series( 1/(3*exp(-x) -2) , x, n+1)*n!, x, n), n = 0..30); # G. C. Greubel, Jun 08 2020

Table[Sum[(-1)^(n-k)*3^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[Exp[x]/(3-2Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 16 2025 *)

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

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

{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, (-1)^(n-k)*3^k*Stirling2(n, k)*k!)}

[sum( (-1)^(n-j)*3^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 08 2020

O.g.f.: A(x) = Sum_{n>=0} n! * 3^n*x^n / Product_{k=0..n} (1+k*x).
O.g.f.: A(x) = 1/(1 - 3*x/(1-2*x/(1 - 6*x/(1-4*x/(1 - 9*x/(1-6*x/(1 - 12*x/(1-8*x/(1 - 15*x/(1-10*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * Stirling2(n,k) * k!.
a(n) = 3*A050351(n) for n>0.
a(n) = Sum_{k=0..n} A123125(n,k)*3^k*2^(n-k). - Philippe Deléham, Nov 30 2011
a(n) ~ n! / (2*log(3/2)^(n+1)). - Vaclav Kotesovec, Jun 13 2013
a(n) = log(3/2) * Integral_{x = 0..oo} (ceiling(x))^n * (3/2)^(-x) dx. - Peter Bala, Feb 06 2015
a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(0) = 1; a(n) = -3*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3*a(n-1) + 2*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
a(n) = (3/2)*A004123(n+1) - (1/2)*0^n. - Seiichi Manyama, Dec 21 2023

cp's OEIS Frontend

A088222 Coefficient of x^n in g.f.^n is A004123(n).

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A167141 G.f.: Sum_{n>=0} A004123(n)^2*log(1+x)^n/n! where 1/(1-2x) = Sum_{n>=0} A004123(n)*log(1+x)^n/n!.

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A174278 Partial sums of A004123.

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A088794 Coefficient of x^n in A(x)^(2n) is A004123(n); self-convolution is A088222.

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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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A050351 Number of 3-level labeled linear rooted trees with n leaves.

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A094417 Generalized ordered Bell numbers Bo(4,n).

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A094416 Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).

A167141 G.f.: Sum_{n>=0} A004123(n)^2log(1+x)^n/n! where 1/(1-2x) = Sum_{n>=0} A004123(n)log(1+x)^n/n!.

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.

A305404 Expansion of Sum_{k>=0} (2k - 1)!!x^k/Product_{j=1..k} (1 - j*x).