A075729 -id:A075729 - cp's OEIS frontend

A034691 Euler transform of powers of 2 [1,2,4,8,16,...].

1, 1, 3, 7, 18, 42, 104, 244, 585, 1373, 3233, 7533, 17547, 40591, 93711, 215379, 493735, 1127979, 2570519, 5841443, 13243599, 29953851, 67604035, 152258271, 342253980, 767895424, 1719854346, 3845443858

Offset: 0

N. J. A. Sloane

nonn

This is the number of different hierarchical orderings that can be formed from n unlabeled elements: these are divided into groups and the elements in each group are then arranged in an "unlabeled preferential arrangement" or "composition" as in A000079. - Thomas Wieder and N. J. A. Sloane, Jun 10 2003
From Gus Wiseman, Mar 03 2016: (Start)
The original Sloane-Wieder definition, "To obtain a hierarchical ordering we partition the elements into unlabeled nonempty subsets and form a composition of each subset," [arXiv:math/0307064] has two possible meanings. The first possible meaning is that we should (1) choose a set partition pi of {1...n} and (2) for each block of pi choose a composition of the number of elements. In this case the correct number of such structures would evidently be counted by A004211 which differs from a(n) for n > 2.
The other possible meaning is that after we have done (1) and (2) above we (3) "forget" the choice of pi. We will have produced a collection M of multisets of compositions. The span of M (its set of distinct elements) is correctly counted by A034691 and it seems that non-isomorphic hierarchical orderings of unlabeled sets are nothing more than multisets of compositions. This discovery is due to Wieder. (End)
The asymptotic formula in the article by N. J. A. Sloane and Thomas Wieder, "The Number of Hierarchical Orderings" (Theorem 3) is incorrect (a multiplicative factor of 1.397... is missing, see my formula below). - Vaclav Kotesovec, Sep 08 2014
Number of partitions of n into 1 sort of 1, 2 sorts of 2's, 4 sorts of 3's, ..., 2^(k-1) sorts of k's, ... . - Joerg Arndt, Sep 09 2014
Also number of normal multiset partitions of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Mar 03 2016

			The normal multiset partitions for a(4) = 18: {{1111},{1222},{1122},{1112},{1233},{1223},{1123},{1234},{1,111},{1,122},{1,112},{1,123},{11,11},{11,12},{12,12},{1,1,11},{1,1,12},{1,1,1,1}}

Vaclav Kotesovec, Table of n, a(n) for n = 0..3190 (first 300 terms from T. D. Noe)
Vaclav Kotesovec, Asymptotics of sequence A034691.
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
Thomas Wieder, An explicit formula for the n-th term.
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [From _Thomas Wieder_, Nov 14 2009]

Cf. A034899, A075729, A247003, A004211, A104500 (Euler transform), A290222 (Multiset transform).

oo := 101: mul( 1/(1-x^j)^(2^(j-1)),j=1..oo): series(%,x,oo): t1 := seriestolist(%); A034691 := n-> t1[n+1];
with(combstruct); SetSeqSetU := [T, {T=Set(S), S=Sequence(U,card >= 1), U=Set(Z,card >=1)},unlabeled]; seq(count(SetSeqSetU,size=j),j=1..12);
# Alternative, uses EulerTransform from A358369:
a := EulerTransform(BinaryRecurrenceSequence(2, 0)):
seq(a(n), n = 0..27); # Peter Luschny, Nov 17 2022

nn = 30; b = Table[2^n, {n, 0, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}],  x] (* T. D. Noe, Nov 21 2011 *)
Table[SeriesCoefficient[E^(Sum[x^k/(1 - 2*x^k)/k, {k, 1, n}]), {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 08 2014 *)
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
allnmsp[0]={};allnmsp[1]={{{1}}};allnmsp[n_Integer]:=allnmsp[n]=Join[allnmsp[n-1],List/@allnorm[n],Join@@Function[ptn,Append[ptn,#]&/@Select[allnorm[n-Length[Join@@ptn]],OrderedQ[{Last[ptn],#}]&]]/@allnmsp[n-1]];
Apply[SequenceForm,Select[allnmsp[4],Length[Join@@#]===4&],{2}] (* to construct the example *)
Table[Length[Complement[allnmsp[n],allnmsp[n-1]]],{n,1,8}] (* Gus Wiseman, Mar 03 2016 *)

A034691(n,l=1+O('x^(n+1)))={polcoeff(1/prod(k=1,n,(l-'x^k)^2^(k-1)),n)} \\ Michael Somos, Nov 21 2011, edited by M. F. Hasler, Jul 24 2017

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(2, 0)
b = EulerTransform(a)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

G.f.: 1 / Product_{n>=1} (1-x^n)^(2^(n-1)).
Recurrence: a(n) = (1/n) * Sum_{m=1..n} a(n-m)*c(m) where c(m) = A083413(m).
a(n) ~ c * 2^n * exp(sqrt(2*n)) / (sqrt(2*Pi) * exp(1/4) * 2^(3/4) * n^(3/4)), where c = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... (see A247003). - Vaclav Kotesovec, Sep 08 2014

A256893 Exponential Riordan array [1, 1/(2-e^x)-1].

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 13, 9, 1, 0, 75, 79, 18, 1, 0, 541, 765, 265, 30, 1, 0, 4683, 8311, 3870, 665, 45, 1, 0, 47293, 100989, 59101, 13650, 1400, 63, 1, 0, 545835, 1362439, 960498, 278901, 38430, 2618, 84, 1, 0, 7087261, 20246445, 16700545, 5844510, 1012431, 92610, 4494, 108, 1

Offset: 0

Peter Luschny, Apr 17 2015

This is also the matrix product of the Stirling set numbers and the unsigned Lah numbers.

This is also the Bell transform of A000670(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			Number triangle starts:
  1;
  0,    1;
  0,    3,     1;
  0,   13,     9,     1;
  0,   75,    79,    18,    1;
  0,  541,   765,   265,   30,   1;
  ...

Cf. A088729 which is a variant based on an (1,1)-offset of the number triangles.
Cf. A131222 which is the matrix product of the unsigned Lah numbers and the Stirling cycle numbers.
Cf. A000670, A075729.

T:= (n, k)-> n!*coeff(series((1/(2-exp(x))-1)^k/k!, x, n+1), x, n):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 17 2015
# The function BellMatrix is defined in A264428.
BellMatrix(n -> polylog(-n-1, 1/2)/2, 9); # Peter Luschny, Jan 29 2016

T[n_, k_] := n!*SeriesCoefficient[(1/(2 - Exp[x]) - 1)^k/k!, {x, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 23 2016, after Alois P. Heinz *)
(* The function BellMatrix is defined in A264428. *)
BellMatrix[PolyLog[-#-1, 1/2]/2&, 9] (* Jean-François Alcover, May 23 2016, after Peter Luschny *)
RiordanArray[d_, h_, n_] := RiordanArray[d, h, n, False];
RiordanArray[d_Function|d_Symbol, h_Function|h_Symbol, n_, exp_:(True | False)] := Module[{M, td, th, k, m},
M[, ] = 0;
td = PadRight[CoefficientList[d[x] + O[x]^n, x], n];
th = PadRight[CoefficientList[h[x] + O[x]^n, x], n];
For[k = 0, k <= n - 1, k++, M[k, 0] = td[[k + 1]]];
For[k = 1, k <= n - 1, k++,
  For[m = k, m <= n - 1, m++,
    M[m, k] = Sum[M[j, k - 1]*th[[m - j + 1]], {j, k - 1, m - 1}]]];
If[exp,
  u = 1;
  For[k = 1, k <= n - 1, k++,
    u *= k;
    For[m = 0, m <= k, m++,
      j = If[m == 0, u, j/m];
      M[k, m] *= j]]];
Table[M[m, k], {m, 0, n - 1}, {k, 0, m}]];
RiordanArray[1&, 1/(2 - Exp[#])-1&, 10, True] // Flatten (* Jean-François Alcover, Jul 16 2019, after Sage program *)

def riordan_array(d, h, n, exp=false):
    def taylor_list(f,n):
        t = SR(f).taylor(x,0,n-1).list()
        return t + [0]*(n-len(t))
    td = taylor_list(d,n)
    th = taylor_list(h,n)
    M = matrix(QQ,n,n)
    for k in (0..n-1): M[k,0] = td[k]
    for k in (1..n-1):
        for m in (k..n-1):
            M[m,k] = add(M[j,k-1]*th[m-j] for j in (k-1..m-1))
    if exp:
        u = 1
        for k in (1..n-1):
            u *= k
            for m in (0..k):
                j = u if m==0 else j/m
                M[k,m] *= j
    return M
riordan_array(1, 1/(2-exp(x)) - 1, 8, exp=true)
# As a matrix product:
def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, stirling_number2)*matrix(ZZ, 8, Lah)

Row sums are given by A075729.
T(n,1) = A000670(n) for n>=1.
T(n,k) = n!/k! * [x^n] (1/(2-exp(x))-1)^k. - Alois P. Heinz, Apr 17 2015

A004211 Shifts one place left under 2nd-order binomial transform.

Original entry on oeis.org

1, 1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, 5120905441, 56878092067, 664920021819, 8155340557697, 104652541401025, 1401572711758403, 19546873773314571, 283314887789276721, 4259997696504874817, 66341623494636864963

Offset: 0

N. J. A. Sloane

Equals the eigensequence of A038207, the square of Pascal's triangle. - Gary W. Adamson, Apr 10 2009
The g.f. of the second binomial transform is 1/(1-2x-x/(1-2x/(1-2x-x/(1-4x/(1-2x-x/(1-6x/(1-2x-x/(1-8x/(1-... (continued fraction). - Paul Barry, Dec 04 2009
Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+2 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=2, otherwise F(k+1)=F(k); see example and Fxtbook link. - Joerg Arndt, Apr 30 2011
It appears that the infinite set of "Shifts 1 place left under N-th order binomial transform" sequences has a production matrix of:
  1, N, 0, 0, 0, ...
  1, 1, N, 0, 0, ...
  1, 2, 1, N, 0, ...
  1, 3, 3, 1, N, ...
  ... (where a diagonal of (N,N,N,...) is appended to the right of Pascal's triangle). a(n) in each sequence is the upper left term of M^n such that N=1 generates A000110, then (N=2 - A004211), (N=3 - A004212), (N=4 - A004213), (N=5 - A005011), ... - Gary W. Adamson, Jul 29 2011
Number of "unlabeled" hierarchical orderings on set partitions of {1..n}, see comments on A034691. - Gus Wiseman, Mar 03 2016
From Lorenzo Sauras Altuzarra, Jun 17 2022: (Start)
Number of n-variate noncontradictory conjunctions of logical equalities of literals (modulo logical equivalence).
Equivalently, number of n-variate noncontradictory Krom formulas with palindromic truth-vector (modulo logical equivalence).
a(n) <= A109457(n). (End)

			From _Joerg Arndt_, Apr 30 2011: (Start)
Restricted growth strings: a(0)=1 corresponds to the empty string;
a(1)=1 to [0]; a(2)=3 to [00], [01], and [02]; a(3) = 11 to
        RGS           F
[ 1]  [ 0 0 0 ]    [ 0 0 0 ]
[ 2]  [ 0 0 1 ]    [ 0 0 0 ]
[ 3]  [ 0 0 2 ]    [ 0 0 2 ]
[ 4]  [ 0 1 0 ]    [ 0 0 0 ]
[ 5]  [ 0 1 1 ]    [ 0 0 0 ]
[ 6]  [ 0 1 2 ]    [ 0 0 2 ]
[ 7]  [ 0 2 0 ]    [ 0 2 2 ]
[ 8]  [ 0 2 1 ]    [ 0 2 2 ]
[ 9]  [ 0 2 2 ]    [ 0 2 2 ]
[10]  [ 0 2 3 ]    [ 0 2 2 ]
[11]  [ 0 2 4 ]    [ 0 2 4 ]. (End)
From _Lorenzo Sauras Altuzarra_, Jun 17 2022: (Start)
The 11 trivariate noncontradictory conjunctions of logical equalities of literals are (x <-> y) /\ (y <-> z), (~ x <-> y) /\ (y <-> z), (x <-> ~ y) /\ (~ y <-> z), (x <-> y) /\ (y <-> ~ z), (x <-> y) /\ (z <-> z), (~ x <-> y) /\ (z <-> z), (x <-> z) /\ (y <-> y), (~ x <-> z) /\ (y <-> y), (y <-> z) /\ (x <-> x), (~ y <-> z) /\ (x <-> x), and (x <-> x) /\ (y <-> y) /\ (z <-> z) (modulo logical equivalence).
The third complete Bell polynomial is x^3 + 3 x y + z; and note that (2^0)^3 + 3*2^0*2^1 + 2^2 = 11.
The truth-vector of (x <-> y) /\ (y <-> z), 10000001, is palindromic. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.5, pp. 366-368
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, On the Limiting Vacillating Tableaux for Integer Sequences, arXiv:2208.13091 [math.CO], 2022.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 22, 29.
David Callan (Proposer), Problem 11567, Amer. Math. Monthly, 118 (2011), 371-378.
Agustina Czenky, Unoriented 2-dimensional TQFTs and the category Rep(S_t wr Z_2), arXiv:2306.08826 [math.QA], 2023. See p. 40.
I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121v3 [math.CO], 2001.
Adam M. Goyt and Lara K. Pudwell, Solution to Monthly Problem 11567, 14 May 2011.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 20.
Janusz Milek, Quantum Implementation of Risk Analysis-relevant Copulas, arXiv:2002.07389 [stat.ME], 2020.
N. J. A. Sloane, Transforms

Cf. A075497 (row sums).
Cf. A038207.
Cf. A000110 (RGS where s(k) <= F(k) + 1), A004212 (RGS where s(k) <= F(k) + 3), A004213 (s(k) <= F(k) + 4), A005011 (s(k) <= F(k) + 5), A005012 (s(k) <= F(k) + 6), A075506 (s(k) <= F(k) + 7), A075507 (s(k) <= F(k) + 8), A075508 (s(k) <= F(k) + 9), A075509 (s(k) <= F(k) + 10).
Cf. A000587, A009235, A317996, A318179 - A318181.
Main diagonal of A261275.
Cf. A034691, A075729, A109457.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*2^(j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 30 2021
# second Maple program:
a:= n -> CompleteBellB(n, seq(2^k, k=0..n)):
seq(a(n), n=0..23);  # Lorenzo Sauras Altuzarra, Jun 17 2022

Table[ Sum[ StirlingS2[ n, k ] 2^(-k+n), {k, n} ], {n, 16} ] (* Wouter Meeussen *)
Table[2^n BellB[n, 1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n):=if n=0 then 1 else sum(2^(n-k)*binomial(n-1,k-1)*a(k-1),k,1,n); /* Vladimir Kruchinin, Nov 28 2011 */

x='x+O('x^66);
egf=exp(intformal(exp(2*x))); /* = 1 + x + 3/2*x^2 + 11/6*x^3 + ... */
/* egf=exp(1/2*(exp(2*x)-1)) */ /* alternative e.g.f. */
Vec(serlaplace(egf))  /* Joerg Arndt, Apr 30 2011 */

def A004211(n): return sum(2^(n-k)*stirling_number2(n, k) for k in (0..n))
print([A004211(n) for n in range(21)]) # Peter Luschny, Apr 15 2020

E.g.f. A(x) satisfies A'(x)/A(x) = e^(2x).
E.g.f.: exp(sinh(x)*exp(x)) = exp(Integral_{t = 0..x} exp(2*t)) = exp((exp(2*x)-1)/2). - Joerg Arndt, Apr 30 2011 and May 13 2011
a(n) = Sum_{k=0..n} 2^(n-k)*Stirling2(n, k). - Emeric Deutsch, Feb 11 2002
G.f.: Sum_{k >= 0} x^k/Product_{j = 1..k} (1-2*j*x). - Ralf Stephan, Apr 18 2004
Stirling transform of A000085. - Vladeta Jovovic May 14 2004
O.g.f.: A(x) = 1/(1-x-2*x^2/(1-3*x-4*x^2/(1-5*x-6*x^2/(1-... -(2*n-1)*x-2*n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n) = e^(-1/2)*2^{n-1}*f_n(1/2). - Milan Janjic, May 30 2008
G.f.: 1/(1-x/(1-2x/(1-x/(1-4x/(1-x/(1-6x/(1-x/(1-8x/(1-... (continued fraction). - Paul Barry, Dec 04 2009
a(n) = upper left term in M^n, M = an infinite square production matrix with an appended diagonal of (2,2,2,...) to the right of Pascal's triangle:
  1, 2, 0, 0, 0, ...
  1, 1, 2, 0, 0, ...
  1, 2, 1, 2, 0, ...
  1, 3, 3, 1, 2, ...
  ... - Gary W. Adamson, Jul 29 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+2*x)*d/dx. Cf. A000110. - Peter Bala, Nov 25 2011
G.f. A(x) satisfies A(x)=1+x/(1-2*x)*A(x/(1-2*x)), a(n) = Sum_{k=1..n} binomial(n-1,k-1)*2^(n-k)*a(k-1), a(0)=1. - Vladimir Kruchinin, Nov 28 2011 [corrected by Ilya Gutkovskiy, May 02 2019]
From Peter Bala, May 16 2012: (Start)
Recurrence equation: a(n+1) = Sum_{k = 0..n} 2^(n-k)*C(n,k)*a(k).
Written umbrally this is a(n+1) = (a + 2)^n (expand the binomial and replace a^k with a(k)). More generally, a*f(a) = f(a+2) holds umbrally for any polynomial f(x). An inductive argument then establishes the umbral recurrence a*(a-2)*(a-4)*...*(a-2*(n-1)) = 1 with a(0) = 1. Compare with the Bell numbers B(n) = A000110(n), which satisfy the umbral recurrence B*(B-1)*...*(B-(n-1)) = 1 with B(0) = 1. Cf. A009235.
Touchard's congruence holds: for odd prime p, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) == 2 (mod p) for odd prime p. (End)
G.f.: (2/E(0) - 1)/x  where E(k) = 1 + 1/(1 + 2*x/(1 - 4*(k+1)*x/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: (1/E(0)-1)/x  where E(k) = 1 - x/(1 + 2*x - 2*x*(k+1)/E(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Sep 21 2012
a(n) = -1 + 2*Sum_{k=0..n} C(n,k)*A166922(k). - Peter Luschny, Nov 01 2012
G.f.: G(0)- 1/x where G(k) = 1 - (4*x*k-1)/(x - x^4/(x^3 - (4*x*k-1)*(4*x*k+2*x-1)*(4*x*k+4*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 08 2013.
G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1-2*k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: -G(0) where G(k) = 1 + 2*(1-k*x)/(2*k*x - 1 - x*(2*k*x - 1)/(x - 2*(1-k*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 29 2013
G.f.: 1/Q(0), where Q(k) = 1 - x/(1 - 2*x*(2*k+1)/( 1 - x/(1 - 4*x*(k+1)/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Apr 15 2013
G.f.: 1 + x/Q(0), where Q(k)= 1 - x*(2*k+3) - x^2*(2*k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 05 2013
For n > 0, a(n) = exp(-1/2)*Sum_{k > 0} (2*k)^n/(k!*2^k). - Vladimir Reshetnikov, May 09 2013
G.f.: -(1+(2*x+1)/G(0))/x, where G(k)= 2*x*k - x - 1 - 2*(k+1)*x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 20 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
Sum_{k=0..n} C(n,k)*a(k)*a(n-k) = 2^n*Bell(n) = A055882(n). - Vaclav Kotesovec, Apr 03 2016
a(n) ~ 2^n * n^n * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^n). - Vaclav Kotesovec, Jan 07 2019, simplified Oct 01 2022
a(n) = B_n(2^0, ..., 2^(n - 1)), where B_n(x_1, ..., x_n) is the n-th complete Bell polynomial. - Lorenzo Sauras Altuzarra, Jun 17 2022

A098407 Number of different hierarchical orderings that can be formed from n unlabeled elements with no repetition of subhierarchies.

Original entry on oeis.org

1, 1, 2, 6, 13, 33, 78, 186, 436, 1028, 2394, 5566, 12877, 29689, 68198, 156194, 356599, 811959, 1843956, 4177436, 9442166, 21295934, 47932572, 107677140, 241443980, 540441068, 1207689636, 2694452060, 6002389882, 13351958546, 29659179804, 65794744420, 145768641091

Offset: 0

Thomas Wieder, Sep 07 2004; corrected Sep 09 2004

nonn

a(n) is the number of finite sets of compositions with total sum n. The case of constant sums is A358904, cf. A074854. The case of distinct sums is A304961, ordered A336127. The ordered version (sequences of distinct compositions) is A358907. - Gus Wiseman, Dec 12 2022

			Let a pair of parentheses () indicate a subhierarchy and let square brackets [] denote a set of subhierarchies, that is, a hierarchy (also called a society). Let the ranks be ordered from left to right and separated by a colon; e.g., (2:3) is a subhierarchy with three elements ("individuals") on top and two elements on the bottom rank.
Then the hierarchical ordering for n = 4 is composed of the following sets: [(1:1),(2)]; [(1),(3)]; [(1),(1:1:1)]; [(1),(2:1)]; [(1),(1:2)]; [(4)]; [(2:2)]; [(1:3)]; [(3:1)]; [(1:1:2)]; [(1:2:1)]; [(2:1:1)]; [(1:1:1:1)]; thus a(4) = 13.
For example, the following hierarchy is not allowed: [(1),(1),(1),(1)] because of the repetition of (1).

Alois P. Heinz, Table of n, a(n) for n = 0..3217
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A011782, A075729.
A034691 counts multisets of compositions, ordered A133494.
A261049 counts sets of partitions, ordered A358906.
Cf. A000009, A000219, A001970, A055887, A075900, A218482, A296122, A304961.

main := proc(n::integer) local a, ListOfPartitions, NumberOfPartitions, APartition, APart, ASet, MultipliticityOfAPart, ndxprttn, ndxprt, Term, Produkt; with(combinat): with(ListTools): a := 0; ListOfPartitions := partition(n); NumberOfPartitions := nops(ListOfPartitions); for ndxprttn from 1 to NumberOfPartitions do APartition := ListOfPartitions[ndxprttn]; ASet := convert(APartition,set); Produkt := 1; for ndxprt from 1 to nops(ASet) do APart := op(ndxprt,ASet); MultipliticityOfAPart := Occurrences(APart, APartition); Term := 2^(APart-1); Term := binomial(Term,MultipliticityOfAPart); Produkt := Produkt * Term; # End of do-loop *** ndxprt ***. end do; a := a + Produkt; # End of do-loop *** ndxprttn ***. end do; print("n, a(n):",n,a); end proc;
PartitionList := proc (n, k) # Authors: # Herbert S. Wilf and Joanna Nordlicht, # Source: # Lecture Notes "East Side West Side,..." # University of Pennsylvania, USA, 2002. # Available from http://www.cis.upenn.edu/~wilf/lecnotes.html # Berechnet die Partitionen von n mit k Summanden. local East, West; if n < 1 or k < 1 or n < k then RETURN([]) elif n = 1 then RETURN([[1]]) else if n < 2 or k < 2 or n < k then West := [] else West := map(proc (x) options operator, arrow; [op(x), 1] end proc, PartitionList(n-1, k-1)) end if; if k <= n-k then East := map(proc(y) options operator, arrow; map(proc (x) options operator, arrow; x+1 end proc, y) end proc, PartitionList(n-k, k)) else East := [] end if; RETURN([op(West), op(East)]) end if end proc;
# second Maple program:
series(exp(add((-1)^(j-1)/j*z^j/(1-2*z^j), j=1..40)), z, 40); # Cf. A102866; Vladeta Jovovic, Feb 19 2008
# alternative Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n>1, 0, 1),
      add(b(n-i*j, i-1)*binomial(2^(i-1), j), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, May 22 2018

terms = 32; CoefficientList[Product[(1 + x^k)^(2^(k-1)), {k, 1, terms+1}] + O[x]^(terms+1), x] // Rest (* Jean-François Alcover, Nov 10 2017, after Vladeta Jovovic *)
nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^k*x^k/(k*(1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 08 2018 *)

a(n) = Sum_{ partitions n = s_1 + ... + s_n } Product_{ Set{s_i} } C(2^(s_i - 1), m(s_i)), where the sum runs over all partitions of n, the product runs over the set of parts of a given partition, s_i is the i-th part in the set of parts, C(k, l) denotes the binomial coefficient and m(s_i) is the multiplicity of part s_i in the given partition.
G.f.: Product_{k>=1} (1+x^k)^(2^(k-1)). - Vladeta Jovovic, Feb 19 2008
a(n) ~ 2^n * exp(sqrt(2*n) - 1/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-2)) = -0.207530918644117743551169251314627032059... - Vaclav Kotesovec, Jun 08 2018
Weigh transform of A011782. - Alois P. Heinz, Jun 25 2018

More terms from Alois P. Heinz, Apr 21 2012

a(0)=1 prepended by Alois P. Heinz, May 22 2018

A034899 Euler transform of powers of 2 [ 2,4,8,16,... ].

Original entry on oeis.org

1, 2, 7, 20, 59, 162, 449, 1200, 3194, 8348, 21646, 55480, 141152, 356056, 892284, 2221208, 5497945, 13533858, 33151571, 80826748, 196219393, 474425518, 1142758067, 2742784304, 6561052331, 15645062126, 37194451937, 88174252924, 208463595471, 491585775018

Offset: 0

N. J. A. Sloane

nonn

			From _Geoffrey Critzer_, Mar 07 2012: (Start)
Per comment in A102866, a(n) is also the number of multisets of binary words of total length n.
a(2) = 7 because the multisets are {a,a}, {b,b}, {a,b}, {aa}, {ab}, {ba}, {bb};
a(3) = 20 because the multisets are {a,a,a}, {b,b,b}, {a,a,b}, {a,b,b}, {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb};
where the words within each multiset are separated by commas. (End)

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..3150 (first 900 terms from Alois P. Heinz)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
G. S. Venkatesh and Kurusch Ebrahimi-Fard, A Formal Power Series Approach to Multiplicative Dynamic Feedback, arXiv:2301.04949 [math.OC], 2023.
Thomas Wieder, Additional comments on this sequence

Cf. A034691, the Euler transform of 1, 2, 4, 8, 16, 32, 64, ...
Cf. A075729, A102866.
Column k=2 of A144074.
Row sums of A055375 and of A209406.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(2^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018 ~

series(1/product((1-x^(n))^(2^(n)),n=1..20),x=0,12); (Wieder)
# second Maple program:
with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d*2^d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 02 2011

nn = 20; p = Product[1/(1 - x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)

m=50; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(2^k))) \\ G. C. Greubel, Nov 09 2018

G.f.: 1/Product_{n>0} (1-x^n)^(2^n). - Thomas Wieder, Mar 06 2005
a(n) ~ c^2 * 2^(n-1) * exp(2*sqrt(n) - 1/2) / (sqrt(Pi) * n^(3/4)), where c = A247003 = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... . - Vaclav Kotesovec, Mar 09 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - 2*x^k))). - Ilya Gutkovskiy, Nov 09 2018

More terms from Thomas Wieder, Mar 06 2005

A049611 a(n) = T(n,2), array T as in A049600.

Original entry on oeis.org

0, 1, 4, 13, 38, 104, 272, 688, 1696, 4096, 9728, 22784, 52736, 120832, 274432, 618496, 1384448, 3080192, 6815744, 15007744, 32899072, 71827456, 156237824, 338690048, 731906048, 1577058304, 3388997632, 7264534528, 15535702016

Offset: 0

Clark Kimberling

Refer to A089378 and A075729 for the definition of hierarchies, subhierarchies and one-step transitions. - Thomas Wieder, Feb 28 2004
We may ask for the number of one-step transitions (NOOST) between all unlabeled hierarchies of n elements with the restriction that no subhierarchies are allowed. As an example, consider n = 4 and the hierarchy H1 = [[2,2]] with two elements on level 1 and two on level 2. Starting from H1 the hierarchies [[1, 3]], [[2, 1, 1]], [[1, 2, 1]] can be reached by moving one element only, but [[1, 1, 2]] cannot be reached in a one-step transitition. The solution is n = 1, NOOST = 0; n = 2, NOOST = 1; n = 3, NOOST = 4; n = 4, NOOST = 13; n = 5, NOOST = 38; n = 6, NOOST = 104; n = 7, NOOST = 272; n = 8, NOOST = 688; n = 9, NOOST = 1696. This is sequence A049611. - Thomas Wieder, Feb 28 2004
If X_1,X_2,...,X_n are 2-blocks of a (2n+2)-set X then, for n>=1, a(n+1) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
In each composition (ordered partition) of the integer n, circle the first summand once, circle the second summand twice, etc. a(n) is the total number of circles in all compositions of n (that is, add k*(k+1)/2 for each composition into k parts). Note the O.g.f. is B(A(x)) where A(x)= x/(1-x) and B(x)= x/(1-x)^3.
This is the Riordan transform with the Riordan matrix A097805 (of the associated type) of the triangular number sequence A000217. See a Feb 17 2017 comment on A097805. - Wolfdieter Lang, Feb 17 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Milan Janjic, Two Enumerative Functions
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
S. Kitaev, J. Remmel, p-Ascent Sequences, arXiv:1503.00914 [math.CO], 2015.
Sergey Kitaev, J. B. Remmel, A note on p-Ascent Sequences, Preprint, 2016.
Igor Makhlin, Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties, arXiv:2003.02916 [math.CO], 2020.
Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

a(n+1)= A055252(n, 0), n >= 0. Row sums of triangle A055249.

Cf. A001793, A058396, A075729, A089378, A000217.

CoefficientList[Series[x (1-x)^2/(1-2x)^3,{x,0,40}],x] (* Harvey P. Dale, Sep 24 2013 *)

concat(0, Vec(x*(1-x)^2/(1-2*x)^3+O(x^99))) \\ Charles R Greathouse IV, Jun 12 2015

G.f.: x*(1-x)^2/(1-2*x)^3.
Binomial transform of quarter squares A002620(n+1): a(n) = Sum_{k=0..n} binomial(n, k)*floor((k+1)^2/4). - Paul Barry, May 27 2003
a(n) = 2^(n-4)*(n^2+5*n+2) - 0^n/8. - Paul Barry, Jun 09 2003
a(n+2) = A001787(n+2) + A001788(n). - Creighton Dement, Aug 02 2005
a(n) = Hyper2F1([-n+1, 3], [1], -1) for n>0. - Peter Luschny, Aug 02 2014
a(n) = Sum_{k=0..n-1} Sum_{j=0..n-1} Sum_{i=0..n-1} binomial(n-1, i+j+k). - Yalcin Aktar, Aug 27 2023

A083355 Number of preferential arrangements for the set partitions of the n-set [1,2,3,...,n].

Original entry on oeis.org

1, 1, 4, 23, 175, 1662, 18937, 251729, 3824282, 65361237, 1241218963, 25928015368, 590852003947, 14586471744301, 387798817072596, 11046531316503163, 335640299372252595, 10835556229612637150, 370383732831919278037, 13363914680277923634517

Offset: 0

Thomas Wieder, Jun 11 2003, May 07 2008

nonn

Labeled analog of A055887. See combstruct commands for more precise definition.
Stirling transform of A000670(n) = [1,3,13,75,...] is a(n) = [1,4,23,175,...]. - Michael Somos, Mar 04 2004
Row sums of A232598. So 2*a(n) is the number of formulas in first-order logic that have an n-place predicate, and don't include a negator. - Tilman Piesk, Nov 28 2013

			Let the colon ":" be a separator between two levels. E.g. in {1,2}:{3} the set {1,2} is on the first level, the set {3} is on the second level.
n=2 gives A083355(2)=4 because we have {1,2} {1}{2} {1}:{2} {2}:{1}.
n=3 gives A083355(3)=23 because we have:
  {1,2,3}
  {1,2}{3} {1,2}:{3} {3}:{1,2}
  {1,3}{2} {1,3}:{2} {2}:{1,3}
  {2,3}{1} {2,3}:{1} {1}:{2,3}
  {1}{2}{3}
  {1}:{2}:{3}
  {3}:{1}:{2}
  {2}:{3}:{1}
  {1}:{3}:{2}
  {2}:{1}:{3}
  {3}:{2}:{1}
  {1}{2}:{3} {1}{3}:{2} {2}{3}:{1}
  {1}:{2}{3} {2}:{1}{3} {3}:{1}{2}.
Examples for the unlabeled case A055887:
n=2 gives A055887(2)=3 because {1,1} {{1}:{1}} {2}
n=3 gives A055887(3)=8 because {1,1,1} {{1}:{1,1}} {{1,1}:{1}} {{1}:{1}:{1}} {1,2} {{1}:{2}} {{2}:{1}} {3}.

Alois P. Heinz, Table of n, a(n) for n = 0..406
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, arXiv:quant-ph/0312202, 2003; J. Phys. A 37 (2004), 3475-3487.
N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
Thomas Wieder, Further comments on A083355
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724.

Cf. A055887.

Cf. A000079, A000670, A034691, A055887, A075729, A232598.

with(combstruct); SeqSetSetL := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >= 1)},labeled]; A083355 := n-> count(SeqSetSetL,size=n);
A083355 := proc(n::integer) #with(combinat); local a,i,j; a:=0; for i from 1 to n do for j from 1 to i do a := a + j!*stirling2(i,j)*stirling2(n,i); od; od; print("n, a(n): ",n, a); end proc; # Thomas Wieder
A083355 := proc() local a,k,n; for n from 1 to 12 do a[n]:=0: for k from 1 to n do a[n]:=a[n]+stirling2(n,k)*A000670(k): od: od: print(a[1],a[2],a[3],a[4],a[5],a[6],a[7],a[8],a[9],a[10],a[11],a[12]); end proc; A000670 := proc(n) local Result,k; Result:=0: for k from 1 to n do Result:=Result+stirling2(n,k)*k! od: end proc;

Range[0, 18]!CoefficientList[Series[1/(2 - E^(E^x - 1)), {x, 0, 18}], x] (* Robert G. Wilson v, Jul 13 2004 *)
a[n_] := Sum[StirlingS2[n, k] PolyLog[-k, 1/2]/2, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 30 2016 *)

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

E.g.f.: 1/(2-exp(exp(x)-1)).
Representation as a double infinite series (Dobinski-type formula): a(n) = (1/2)*Sum_{k>=1} (k^n/k!)*Sum_{p>=1} p^k/(2*exp(1))^p, n >= 1. - Karol A. Penson and Pawel Blasiak (blasiak(AT)lptl.jussieu.fr), Nov 30 2003
a(n) ~ n!/(2 * c * (log c)^(n+1)) where c = 1 + log 2.
a(n) = Sum_{k=1..n} C(n, k)*Bell(k)*a(n-k). - Vladeta Jovovic, Jul 24 2003
a(n) = Sum_{i=1..n} Sum_{j=1..i} j!*Stirling2(i,j)*Stirling2(n,i). - Thomas Wieder, May 09 2005
a(n) = Sum_{k=1..n} S2(n,k) A000670(k).
a(n) = Sum_{k >= 0} Bell(n,k)/2^(k+1), where Bell(n,x) = Sum_{k = 0..n} Stirling2(n,k)*x^k denotes the n-th Bell or exponential polynomial. - Peter Bala, Jul 09 2014

A140585 Total number of all hierarchical orderings for all set partitions of n.

Original entry on oeis.org

1, 4, 20, 129, 1012, 9341, 99213, 1191392, 15958404, 235939211, 3817327362, 67103292438, 1273789853650, 25973844914959, 566329335460917, 13150556885604115, 324045146807055210, 8446201774570017379, 232198473069120178475, 6715304449424099384968

Offset: 1

Thomas Wieder, May 17 2008

nonn

			We are considering all set partitions of the n-set {1,2,3,...,n}.
For each such set partition we examine all possible hierarchical arrangements of the subsets. A hierarchy is a distribution of elements (sets in the present case) onto levels.
A distribution onto levels is "hierarchical" if a level L+1 contains at most as many elements as level L. Thus for n=4 the arrangement {1,2}:{3}{4} is not allowed.
Let the colon ":" separate two consecutive levels L and L+1.
n=2 --> 1+3=4
{1,2} {1}{2}
{1}:{2}
{2}:{1}
-----------------------
n=3 --> 1+9+10=20
1*1 3*3=9 1*10
{1,2,3} {1,2}{3} {1}{2}{3}
{1,3}{2}
{2,3}{1} {1}{2}:{3}
{3}{1}:{2}
{1,2}:{3} {2}{3}:{1}
{1,3}:{2}
{2,3}:{1} {1}:{2}:{3}
{3}:{1}:{2}
{3}:{1,2} {2}:{3}:{1}
{2}:{1,3} {1}:{3}:{2}
{1}:{2,3} {2}:{1}:{3}
{3}:{2}:{1}
-----------------------
n=4 --> 1+12+9+60+47=129
1*1 4*3=12 3*3=9 6*10=60 1*47
{1,2,3,4} {1,2,3}{4} {1,2}{3,4} {1,2}{3}{4} {1}{2}{3}{4}
{1,2,4}{3} {1,3}{2,4} {1,2}{3}:{4}
{1,3,4}{2} {1,4}{2,3} {1,2}{4}:{3} {1}{2}:{3}:{4}
{2,3,4}{1} {1}{2}:{3,4} {1}{3}:{2}:{4}
{1,2}:{3,4} {1,2}:{3}:{4} {1}{4}:{2}:{3}
{1,2,3}:{4} {1,3}:{2,4} {1,2}:{4}:{3} {1}{2}:{4}:{3}
{1,2,4}:{3} {1,4}:{2,3} {1}:{2}:{3,4} {1}{3}:{4}:{2}
{1,3,4}:{2} {3,4}:{1,2} {2}:{1}:{3,4} {1}{4}:{3}:{2}
{2,3,4}:{1} {2,4}:{1,3} {1}:{3,4}:{2}
{2,3}:{1,4} {2}:{3,4}:{1} {2}{3}:{1}:{4}
{4}:{1,2,3} {2}{4}:{1}:{3}
{3}:{1,2,4} likewise for: {2}{3}:{4}:{1}
{2}:{1,3,4} {3,4}{1}{2} {2}{4}:{3}:{1}
{1}:{2,3,4} {2,4}{1}{3}
{2,3}{1}{4} {3}{4}:{1}:{2}
{1,4}{2}{3} {3}{4}:{2}:{1}
{1,3}{2}{4}
{1}{2}:{3}{4}
{1}{3}:{2}{4}
{1}{4}:{2}{3}
{2}{3}:{1}{4}
{2}{4}:{1}{3}
{3}{4}:{1}{2}
{2}{3}{4}:{1}
{1}{3}{4}:{2}
{1}{2}{4}:{3}
{1}{2}{3}:{4}
{1}:{2}:{3}:{4}
+23 permutations

Vaclav Kotesovec, Table of n, a(n) for n = 1..420 (first 160 terms from Alois P. Heinz)
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [_Thomas Wieder_, Nov 14 2009]

Cf. A005651, A075729, A034691, A083355, A109186, A000670.

A140585 := proc(n::integer) local k, Result; Result:=0: for k from 1 to n do Result:=Result+stirling2(n,k)*A005651(k); end do; lprint(Result); end proc;
E.g.f.: series(1/mul(1-(exp(x)-1)^k/k!,k=1..10),x=0,10). # Thomas Wieder, Sep 04 2008
# second Maple program:
with(numtheory): b:= proc(k) option remember; add(d/d!^(k/d), d=divisors(k)) end: c:= proc(n) option remember; `if`(n=0, 1, add((n-1)!/ (n-k)!* b(k)* c(n-k), k=1..n)) end: a:= n-> add(Stirling2(n, k) *c(k), k=1..n): seq(a(n), n=1..30); # Alois P. Heinz, Oct 10 2008

Table[n!*SeriesCoefficient[1/Product[(1-(E^x-1)^k/k!),{k,1,n}],{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Sep 03 2014 *)

Stirling transform of A005651 = Sum of multinomial coefficients: a(n) = Sum_{i=1..n} S2(n,k) A005651(k).
E.g.f.: 1/Product_{k>=1} (1 - (exp(x)-1)^k/k!). - Thomas Wieder, Sep 04 2008
a(n) ~ c * n! / (log(2))^n, where c = 1/(2*log(2)) * Product_{k>=2} 1/(1-1/k!) = A247551 / (2*log(2)) = 1.82463230250447246267598544320244231645906135137... . - Vaclav Kotesovec, Sep 04 2014, updated Jan 21 2017

More terms from Alois P. Heinz, Oct 10 2008

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 0..531
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).

E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).

A097237 Number of hierarchical orderings ("societies") of n labeled elements ("individuals") with at least two occupied levels.

Original entry on oeis.org

0, 2, 12, 86, 780, 8462, 106092, 1507046, 23905740, 418581662, 8014481772, 166501716086, 3728936827980, 89530481995502, 2293539506425452, 62429371709206406, 1799021068567370700, 54707449240102350782, 1750530594833378049132, 58787407236482804618006

Offset: 1

Thomas Wieder, Aug 02 2004

nonn

			a(3) = 12. Let : denote the partition of n labeled individuals 1,2,3,4 into x=2 sets (i.e. "societies"). E.g., in 12:34 one has a single society with members 1 and 2 and a further single society with members 3 and 4. Let | denote a higher level (within a single society), e.g., in 1|2 the individual 2 is one level up with respect to individual 1. The order of individuals on a level is insignificant, e.g., 12|34 is equivalent to 21|43. For n = 3 and x = 2 one has 12|3; 23|1; 13|2; 1|23; 2|13; 3|12; 1|2|3; 2|3|1; 3|1|2; 1|3|2; 3|2|1; 2|1|3; which gives 12 different societies with at least 2 occupied levels.

Alois P. Heinz, Table of n, a(n) for n = 1..150
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Cf. A075729, A097236.

with(combstruct); SetSeq2SetL:=[T,{T=Set(S), S=Sequence(U,card>=2), U=Set(Z,card >= 1)},labeled];
# where x is an integer 1, 2, 3,... ; x=2 gives 2 levels per society.
seq (count (SetSeq2SetL,size=j), j=1..12);

Rest[CoefficientList[Series[E^((2*E^x-E^(2*x)-1) / (E^x-2)), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Sep 13 2014 *)

E.g.f.: exp(-(exp(z)^2-2*exp(z)+1)/(-2+exp(z))).

a(n) ~ exp(sqrt(2*n/log(2)) + 1/(4*log(2)) - n - 7/4) * n^(n-1/4) / (2^(3/4) * log(2)^(n+1/4)). - Vaclav Kotesovec, Sep 13 2014

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A004211 Shifts one place left under 2nd-order binomial transform.

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A098407 Number of different hierarchical orderings that can be formed from n unlabeled elements with no repetition of subhierarchies.

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A034899 Euler transform of powers of 2 [ 2,4,8,16,... ].

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A049611 a(n) = T(n,2), array T as in A049600.

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