A053549 -id:A053549 - cp's OEIS frontend

A198046 Exponential transform of A053549.

1, 1, 3, 19, 225, 4841, 192355, 14537643, 2135997537, 616565334097, 351243585487331, 395958973398105283, 885030941975862363649, 3928075680727698371316537, 34658158001445631936261356547, 608435501761943981290097259909211

Offset: 0

Geoffrey Critzer, Oct 20 2011

nonn

a(n) is the number of ways to designate a node in each connected component over all simple labeled graphs on n nodes.

Alois P. Heinz, Table of n, a(n) for n = 0..80

Cf. A053549.

g:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*g(k), k=1..n-1)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1) *j*g(j) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 17 2015

nn=20; a=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}]; Range[0,nn]! CoefficientList[Series[Exp[x D[Log[a],x]], {x,0,nn}], x]

seq(n)={Vec(serlaplace(exp(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))))))} \\ Andrew Howroyd, Jun 18 2018

E.g.f.: exp(A(x)) where A(x) is the e.g.f. for A053549.

a(6), a(10) corrected by Alois P. Heinz, Mar 18 2015

A053550 Reversion of [1,2,12,152,3640,...] (A053549).

Original entry on oeis.org

1, 2, -4, 72, -2168, 114864, -10671216, 1787642272, -557621854080, 332091938264960, -383877081387120640, 870132605833456937472, -3891989736269204588784384, 34485850497928703119235323904, -606809291874393995479640098992128

Offset: 1

N. J. A. Sloane, Jan 16 2000

sign

Index entries for reversions of series

A053551 REVEGF transform of [1,2,12,152,3640,...] (A053549).

Original entry on oeis.org

1, 4, 12, 128, -2360, 225504, -31479056, 7663698176, -3268091071872, 2502155367185920, -3525130171601665792, 9336272433375180957696, -47283253230423958235278336, 463782058088334028985937178624, -8890882429150197015851583349985280

Offset: 1

N. J. A. Sloane, Jan 16 2000

sign

A053552 E.g.f. is obtained by reversion of e.g.f. for A053549.

Original entry on oeis.org

1, -2, 0, -32, -1000, -61104, -6622448, -1274984320, -444075170688, -286507347953920, -349824570713902592, -821921957328576534528, -3760909487461975276739584, -33794780371666462710538704896, -599695123654141294961479035617280

Offset: 1

N. J. A. Sloane, Jan 16 2000

sign

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.20, G^<-1>(x).

Index entries for reversions of series

seriestoseries(G, 'revogf'); # where G is e.g.f. for A053549; i.e. use ordinary reversion on the e.g.f. and treat answer as another e.g.f.

A001187 Number of connected labeled graphs with n nodes.

Original entry on oeis.org

1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592, 66296291072, 34496488594816, 35641657548953344, 73354596206766622208, 301272202649664088951808, 2471648811030443735290891264, 40527680937730480234609755344896, 1328578958335783201008338986845427712

Offset: 0

N. J. A. Sloane

"Based on experimental data obtained using the software LattE [14] and the Online Encyclopedia of Integer Sequences [19], we make the following conjecture: Conjecture 11. For j >= 2, Vol(C_j ) is equal to the number of labeled connected graphs on j - 1 vertices." [Beck et al., 2011]
For n > 1, a(n) is log-convex. Furthermore, a(n+1)*a(n-1) ~ 2*a(n)*a(n). - Ran Pan, Oct 28 2015
a(n) is also the number of tournaments on {1,...,n} for which 1 is reachable from every vertex. - Don Knuth, Aug 06 2020

			E.g.f.: 1 + x + x^2/2! + 4*x^3/3! + 38*x^4/4! + 728*x^5/5! + 26704*x^6/6! + 1866256*x^7/7! + 251548592*x^8/8! + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 398-402.
D. G. Cantor, personal communication.
Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519).
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 518.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 7.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.1.
H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 78.

T. D. Noe, Table of n, a(n) for n = 0..50
Matthias Beck, Benjamin Braun and Nguyen Le, Mahonian partition identities via polyhedral geometry, arXiv:1103.1070 [math.NT], 2011.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, MS Thesis, 2002.
Marco Coraggio and Mario di Bernardo, Data-driven design of complex network structures to promote synchronization, arXiv:2309.10941 [eess.SY], 2023.
Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 138.
Philippe Fournier Viger, Ganghuan He, Chun-Wei Jerry Lin, and Heitor Murilo Gomes, Mining Attribute Evolution Rules in Dynamic Attributed Graphs, Proceedings of the Big Data Analytics and Knowledge Discovery, 22nd International Conference, (Bratislava, Slovakia, DaWaK 2020).
Adriano M. Garsia, James Haglund, Dun Qiu, and Marino Romero, e-Positivity Results and Conjectures, arXiv:1904.07912 [math.CO], 2019.
E. N. Gilbert, Enumeration of labelled graphs, Canad. J. Math., 8 (1956), 405-411.
E. N. Gilbert, Enumeration of labelled graphs (Annotated scanned copy)
Markus Kirchweger and Stefan Szeider, SAT Modulo Symmetries for Graph Generation and Enumeration, ACM Trans. Comput. Logic (2024). See p. 27.
M. Konvalinka and I. Pak, Cayley compositions, partitions, polytopes, and geometric bijections, preprint.
M. Konvalinka and I. Pak, Cayley compositions, partitions, polytopes, and geometric bijections, Journal of Combinatorial Theory, Series A, Volume 123, Issue 1, April 2014, Pages 86-91.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 32.
Arun P. Mani and R. J. Stones, The Number of Labeled Connected Graphs Modulo Prime Powers, SIAM Journal on Discrete Mathematics, Vol. 30, No. 2, pp. 1046-1057.
Alexey A. Melnikov, Leonid E. Fedichkin, Ray-Kuang Lee, and Alexander Alodjants, Machine learning transfer efficiencies for noisy quantum walks, arXiv:2001.05472 [quant-ph], 2020. Also in Advanced Quantum Technologies (2020) Vol. 3, 1900115.
Albert Nijenhuis and Herbert S. Wilf, The enumeration of connected graphs and linked diagrams, J. Combin. Theory Ser. A 27 (1979), no. 3, 356--359. MR0555804 (82b:05074)
J. Novak, Three lectures on free probability, arXiv preprint arXiv:1205.2097 [math.CO], 2012.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
R. W. Robinson, First 50 terms of A1187 and A1188
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Labeled Graph
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 87, Eq. 3.10.2.

Logarithmic transform of A006125 (labeled graphs).
Row sums of triangle A062734.
Cf. A053549.

m:=30;
f:= func< x | 1+Log( (&+[2^Binomial(n,2)*x^n/Factorial(n): n in [0..m+3]]) ) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 04 2022

t1 := 1+log( add(2^binomial(n,2)*x^n/n!,n=0..30)): t2 := series(t1,x,30): A001187 := n->n!*coeff(t2,x,n);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 26 2013
# Alternative:
a := proc(n) option remember;
    2^((n-1)*n/2) - add(binomial(n-1, k)*2^((k-n+1)*(k-n+2)/2)*a(k+1), k=0..n-2)
end:
seq(a(n), n=0..16); # Peter Luschny, Feb 21 2023

m:=20; g = Sum[2^Binomial[n, 2] x^n/n!, {n,0,m}]; Range[0,m]! CoefficientList[Series[Log[g] +1, {x,0,m}], x] (* Geoffrey Critzer, Nov 12 2011 *)
a[n_]:= a[n]= If[n==0, 1, 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]* 2^((n-k)*(n-k-1)/2)*a[k], {k,1,n-1}]/n]; Table[a[n], {n,0,20}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Log[ Sum[2^(k(k-1)/2) x^k/k!, {k,0, n}]], {x, 0, n}]]; (* Michael Somos, Jul 11 2019 *)

{a(n) = if(n<0, 0, n! * polcoeff( 1 + log( sum( k=0, n, 2^binomial(k, 2) * x^k / k!, x * O(x^n))), n))}; /* Michael Somos, Jun 12 2000 */

from functools import lru_cache
import gmpy2
@lru_cache(None)
def A001187(n):
  if n == 0:
    return 1
  s = gmpy2.mpz(0)
  for k in range(1, n):
    s += k * gmpy2.comb(n, k) * 2**((n - k)*(n - k - 1)//2) * A001187(k)
  return 2**(n*(n-1)//2) - s // n # John Reimer Morales, Aug 15 2025

@cached_function
def A001187(n):
    if n == 0: return 1
    return 2^(n*(n-1)/2)- sum(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*A001187(k) for k in (1..n-1))/n
[A001187(n) for n in (0..15)] # Peter Luschny, Jan 17 2016

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

E.g.f.: 1 + log(Sum_{n>=0} 2^binomial(n, 2) * x^n / n!). - Michael Somos, Jun 12 2000

A095983 Number of 2-edge-connected labeled graphs on n nodes.

Original entry on oeis.org

0, 0, 0, 1, 10, 253, 11968, 1047613, 169181040, 51017714393, 29180467201536, 32121680070545657, 68867078000231169536, 290155435185687263172693, 2417761175748567327193407488, 40013922635723692336670167608181, 1318910073755307133701940625759574016

Offset: 0

Yifei Chen (yifei(AT)mit.edu), Jul 17 2004

nonn

From Falk Hüffner, Jun 28 2018: (Start)
Essentially the same sequence arises as the number of connected bridgeless labeled graphs (graphs that are k-edge connected for k >= 2, starting elements of this sequence are 1, 1, 0, 1, 10, 253, 11968, ...).
Labeled version of A007146. (End)
The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. This sequence counts graphs with spanning edge-connectivity >= 2, which, for n > 1, are connected graphs with no bridges. - Gus Wiseman, Sep 20 2019

Jinyuan Wang, Table of n, a(n) for n = 0..82
P. Hanlon and R. W. Robinson, Counting bridgeless graphs, J. Combin. Theory, B 33 (1982), 276-305.
Gus Wiseman, The a(4) = 10 labeled graphs with spanning edge-connectivity >= 2.
Gus Wiseman, Non-isomorphic representatives of the a(5) = 253 graphs with spanning edge-connectivity >= 2.

Cf. A053549, A198046.
The unlabeled version is A007146.
Row sums of A327069 if the first two columns are removed.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Graphs with spanning edge-connectivity 2 are A327146.
Graphs with non-spanning edge-connectivity >= 2 are A327200.
2-vertex-connected graphs are A013922.
Graphs without endpoints are A059167.
Graphs with spanning edge-connectivity 1 are A327071.
Cf. A001187, A002218, A052446, A059166, A100743, A322395, A327079, A327144.

seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[ Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k-1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];
seq[16] (* Jean-François Alcover, Aug 13 2019, after Andrew Howroyd *)
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],spanEdgeConn[Range[n],#]>=2&]],{n,0,5}] (* Gus Wiseman, Sep 20 2019 *)

\\ here p is initially A053549, q is A198046 as e.g.f.s.
seq(n)={my(v=vector(n));
my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))));
my(q=x*exp(p)); p-=q;
for(k=3, n, my(c=polcoeff(p,k)); v[k]=c*(k-1)!; p-=c*q^k);
concat([0],v)} \\ Andrew Howroyd, Jun 18 2018

seq(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))/x-1), -(n+1))} \\ Andrew Howroyd, Dec 28 2020

a(n) = A001187(n) - A327071(n). - Gus Wiseman, Sep 20 2019

Name corrected and more terms from Pavel Irzhavski, Nov 01 2014
Offset corrected by Falk Hüffner, Jun 17 2018
a(12)-a(16) from Andrew Howroyd, Jun 18 2018

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).

A327072 Triangle read by rows where T(n,k) is the number of labeled simple connected graphs with n vertices and exactly k bridges.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 3, 0, 10, 12, 0, 16, 0, 253, 200, 150, 0, 125, 0, 11968, 7680, 3600, 2160, 0, 1296, 0, 1047613, 506856, 190365, 68600, 36015, 0, 16807, 0, 169181040, 58934848, 16353792, 4695040, 1433600, 688128, 0, 262144, 0, 51017714393, 12205506096, 2397804444, 500828832, 121706550, 33067440, 14880348, 0, 4782969, 0

Offset: 0

Gus Wiseman, Aug 24 2019

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).

Warning: In order to be consistent with A001187, we have treated the n = 0 and n = 1 cases in ways that are not consistent with A095983.

			Triangle begins:
    1
    1   0
    0   1   0
    1   0   3   0
   10  12   0  16   0
  253 200 150   0 125   0

Andrew Howroyd, Table of n, a(n) for n = 0..1325
Gus Wiseman, The 10 + 12 + 16 graphs counted in row n = 4.

Column k = 0 is A095983, if we assume A095983(0) = A095983(1) = 1.
Column k = 1 is A327073.
Column k = n - 1 is A000272.
Row sums are A001187.
The unlabeled version is A327077.
Row sums without the first column are A327071.
Cf. A001349, A007146, A052446, A054592, A059166, A322395, A327069, A327148.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[If[n<=1&&k==0,1,Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]==1&&Count[Table[Length[Union@@Delete[#,i]]1,{i,Length[#]}],True]==k&]]],{n,0,4},{k,0,n}]

\\ p is e.g.f. of A053549.
T(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))), v=Vec(1+serreverse(serreverse(log(x/serreverse(x*exp(p))))/exp(x*y+O(x^n))))); vector(#v, k, max(0,k-2)!*Vecrev(v[k], k)) }
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 28 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 28 2020

A177777 E.g.f. satisfies: L(x) = xSum_{n>=0} (1/n!)Product_{k=0..n-1} L(2^k*x).

Original entry on oeis.org

1, 2, 12, 152, 3640, 160224, 13063792, 2012388736, 596666619648, 344964885948160, 392058233038486784, 880255154481199466496, 3916538634445633156373504, 34603083354426212294072477696

Offset: 1

Paul D. Hanna, May 19 2010

nonn

An analog of the LambertW function.

A053549 without the leading term. - R. J. Mathar, May 24 2010

			E.g.f.: L(x) = x + 2*x^2/2! + 12*x^3/3! + 152*x^4/4! + 3640*x^5/5! +...
which is invariant under the series:
L(x)/x = 1 + L(x) + L(x)L(2x)/2! + L(x)L(2x)L(4x)/3! + L(x)L(2x)L(4x)L(8x)/4! +...
Let B(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 64*x^4/4! + 1024*x^5/5! +...
so that log(B(x)) = x + x^2/2! + 4*x^3/3! + 38*x^4/4! + 728*x^5/5! +...+ A001187(n)*x^n/n! +...
then L(x) = x*d/dx log(B(x)) which also satisfies:
1/B(x) = 1 - L(x) + L(x)L(2x)/2! - L(x)L(2x)L(4x)/3! + L(x)L(2x)L(4x)L(8x)/4! -+...

Vaclav Kotesovec, Table of n, a(n) for n = 1..70

{a(n,r=1)=local(A=x+x^2);for(i=1,n,A=x*sum(m=0,n,r^m/m!*prod(k=0,m-1,subst(A,x,2^k*x+x*O(x^n)))));n!*polcoeff(A,n)}

a(n) = n*A001187(n), where A001187(n) is the number of connected labeled graphs with n nodes.
Let B(x) = Sum_{n>=0} 2^(n(n-1)/2)*x^n/n! then
. L(x) = x*d/dx log(B(x)) = x*B'(x)/B(x) and
. 1/B(x) = Sum_{n>=0} (-1)^n/n!*Product_{k=0..n-1} L(2^k*x).

A228315 Triangular array read by rows: T(n,k) is the number of rooted labeled simple graphs on {1,2,...,n} such that the root is in a component of size k; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 2, 6, 6, 12, 32, 24, 48, 152, 320, 160, 240, 760, 3640, 6144, 1920, 1920, 4560, 21840, 160224, 229376, 43008, 26880, 42560, 152880, 1121568, 13063792, 16777216, 1835008, 688128, 680960, 1630720, 8972544, 104510336, 2012388736

Offset: 1

Geoffrey Critzer, Aug 26 2013

Row sums = A095340.
Column 1 = A123903.
T(n,k)   = A223894(n,k)*k.
Diagonal = A053549.

			1;
2,    2;
6,    6,    12;
32,   24,   48,    152;
320,  160,  240,   760,    3640;

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7.

Alois P. Heinz, Rows n = 1..45, flattened
Index entries for sequences related to graphs

Cf. A070166.

b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
    end:
T:= (n, k)-> binomial(n, k)*k*b(k)*2^((n-k)*(n-k-1)/2):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 26 2013

nn = 10; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; a =
Drop[Range[0, nn]! CoefficientList[Series[Log[g], {x, 0, nn}], x],
  1]; Table[
  Table[Binomial[n, k] k a[[k]] 2^Binomial[n - k, 2], {k, 1, n}], {n,
   1, 7}] // Grid

T(n,k) = binomial(n,k)*k*A001187(k)*A006125(n-k).

cp's OEIS Frontend

A198046 Exponential transform of A053549.

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A053550 Reversion of [1,2,12,152,3640,...] (A053549).

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A053551 REVEGF transform of [1,2,12,152,3640,...] (A053549).

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A053552 E.g.f. is obtained by reversion of e.g.f. for A053549.

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A001187 Number of connected labeled graphs with n nodes.

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A095983 Number of 2-edge-connected labeled graphs on n nodes.

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A274804 The exponential transform of sigma(n).

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A327072 Triangle read by rows where T(n,k) is the number of labeled simple connected graphs with n vertices and exactly k bridges.

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A177777 E.g.f. satisfies: L(x) = xSum_{n>=0} (1/n!)Product_{k=0..n-1} L(2^k*x).