A035051 -id:A035051 - cp's OEIS frontend

A030438 a(n) = A030019(n)n! (or A035051(n-1)!).

1, 1, 2, 24, 696, 37320, 3201840, 401914800, 69458497920, 15813882201600, 4587474713068800, 1651825133370720000, 722868238335090355200, 377862727500237858278400, 232536825223980698118297600

Offset: 0

David Warme (warme(AT)s3i.com)

nonn

Warren D. Smith and David Warme, Paper in preparation, 2002.

Index entries for sequences related to rooted trees

Cf. A030019, A035051.

a(n) = (n-1)! Sum_{i=0}^{n-1} Stirling2(n-1, i) n^i, n >= 1.

A030019 Number of labeled spanning trees in the complete hypergraph on n vertices (all hyperedges having cardinality 2 or greater).

Original entry on oeis.org

1, 1, 1, 4, 29, 311, 4447, 79745, 1722681, 43578820, 1264185051, 41381702275, 1509114454597, 60681141052273, 2667370764248023, 127258109992533616, 6549338612837162225, 361680134713529977507, 21333858798449021030515, 1338681172839439064846881

Offset: 0

David Warme (warme(AT)s3i.com)

Equivalently, this is the number of "hypertrees" on n labeled nodes, i.e. connected hypergraphs that have no cycles, assuming that each edge contains at least two vertices. - Don Knuth, Jan 26 2008. See A134954 for hyperforests.
Also number of labeled connected graphs where every block is a complete graph (cf. A035053).
Let H = (V,E) be the complete hypergraph on N labeled vertices (all edges having cardinality 2 or greater). Let e in E and K = |e|. Then the number of distinct spanning trees of H that contain edge e is g(N,K) = K * E[X_N^{N-K}] / N and the K=1 case gives this sequence. Clearly there is some deep structural connection between spanning trees in hypergraphs and Poisson moments.

Warren D. Smith and David Warme, Paper in preparation, 2002.

Alois P. Heinz, Table of n, a(n) for n = 0..370 (first 101 terms from T. D. Noe)
Ayomikun Adeniran and Catherine Yan, Gončarov Polynomials in Partition Lattices and Exponential Families, arXiv:1907.07814 [math.CO], 2019.
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 810.
Louis H. Kalikow, Enumeration of parking functions, allowable permutation pairs, and labeled trees, PhD thesis, Brandeis University, 1999.
R. Lorentz, S. Tringali, and C.H. Yan, Generalized Goncarov polynomials, arXiv preprint arXiv:1511.04039, 2015.
Adam Piggott, The symmetries of Mccullough-Miller space, 2011, preprint.
Adam Piggott, The symmetries of Mccullough-Miller space, Algebra and Discrete Mathematics 14(2) (2012), 239-266.
D. M. Warme, Spanning Trees in Hypergraphs with Applications to Steiner Trees, PhD thesis, University of Virginia, 1998, Table 5.1.
D. M. Warme, Spanning Trees in Hypergraphs with Applications to Steiner Trees, PhD thesis, University of Virginia, 1998, Table 5.1.
Index entries for sequences related to trees

Cf. A030438, A035051, A035053, A134954, A134956, A134958. A052888, A210587.

a[n_] := Sum[ StirlingS2[n-1, i]*n^(i-1), {i, 0, n-1}]; a[0] = 1; Table[a[n], {n, 0, 18}](* Jean-François Alcover, Sep 12 2012, from 2nd formula *)

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

/* E.g.f. of sequence shifted left one place: */
{a(n)=local(A=1+x); for(i=1, n, A=exp(-1)*sum(m=0, 2*n+10, exp(m*x*A+x*O(x^n))/m!)); round(n!*polcoeff(A, n))} /* Paul D. Hanna */

a(n) = A035051(n)/n for n > 0.
a(n) = Sum_{i=0...n-1} Stirling2(n-1, i) n^(i-1), n >= 1. (Warme, Corollary 3.15.1, p. 59)
a(n) = E[X_n^{n-1}] / n, n >= 1, where X_n is a Poisson random variable with mean n.
1 = Sum_{n>=0} a(n+1) * x^n/n! * exp( -(n+1)*(exp(x)-1) ). - Paul D. Hanna, Jun 11 2011
E.g.f. satisfies: A(x) = Sum_{n>=0} exp(n*x*A(x)-1)/n! = Sum_{n>=0} a(n+1)*x^n/n!. - Paul D. Hanna, Sep 25 2011
Dobinski-type formula: a(n) = 1/e^n*sum {k = 0..inf} n^(k-1)*k^(n-1)/k!. Cf. A052888. For a refinement of this sequence see A210587. - Peter Bala, Apr 05 2012
a(n) ~ n^(n-2) / (sqrt(1+LambertW(1)) * (LambertW(1))^(n-1) * exp((2-1/LambertW(1))*n)). - Vaclav Kotesovec, Jul 26 2014

More terms, formula and comment from Christian G. Bower Dec 15 1999

A035053 Number of connected graphs on n unlabeled nodes where every block is a complete graph.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 22, 59, 165, 496, 1540, 4960, 16390, 55408, 190572, 665699, 2354932, 8424025, 30424768, 110823984, 406734060, 1502876903, 5586976572, 20884546416, 78460794158, 296124542120, 1122346648913, 4270387848473

Offset: 0

Christian G. Bower, Oct 15 1998

Equivalently, this is the number of "hypertrees" on n unlabeled nodes, i.e., connected hypergraphs that have no cycles, assuming that each edge contains at least two vertices. - Don Knuth, Jan 26 2008. See A134955 for hyperforests.

Graphs where every block is a complete graph are also called block graphs or clique tree. They can be characterized as induced-diamond-free chordal graphs. - Falk Hüffner, Jul 25 2019

			From _Gus Wiseman_, May 20 2018: (Start)
Non-isomorphic representatives of the a(5) = 9 hypertrees are the following:
  {{1,2,3,4,5}}
  {{1,5},{2,3,4,5}}
  {{1,2,5},{3,4,5}}
  {{1,2},{2,5},{3,4,5}}
  {{1,4},{2,5},{3,4,5}}
  {{1,5},{2,5},{3,4,5}}
  {{1,3},{2,4},{3,5},{4,5}}
  {{1,4},{2,5},{3,5},{4,5}}
  {{1,5},{2,5},{3,5},{4,5}}
(End)

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 71, (3.4.14).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
Robert Hellmann and Eckard Bich, A systematic formulation of the virial expansion for nonadditive interaction potentials, J. Chem. Phys. 135, 084117 (2011); doi:10.1063/1.3626524 (7 pages).
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Block Graph
Eric Weisstein's World of Mathematics, Connected Graph
Wikipedia, Block graph.

Cf. A007549, A007563, A007716, A030019, A035051, A035052, A054921, A134957, A134959, A245566, A304867, A304887, A304937.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: b:= etr(B): c:= etr(b): B:= n-> if n=0 then 0 else c(n-1) fi: C:= etr(B): a:= n-> B(n)+C(n) -add(B(k)*C(n-k), k=0..n): seq(a(n), n=0..30); # Alois P. Heinz, Sep 09 2008

ClearAll[etr, b, a]; etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; b[0]=0; b[n_] := b[n] = etr[etr[b]][n-1]; a[n_] := b[n] + etr[b][n] - Sum[b[k]*etr[b][n-k], {k, 0, n}]; Table[ a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 09 2012, after Alois P. Heinz *)

\\ here b(n) is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u)))} \\ Andrew Howroyd, May 22 2018

G.f.: A(x)=1+(C(x)-1)*(1-B(x)). B: G.f. for A007563. C: G.f. for A035052.
a(n) ~ c * d^n / n^(5/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.245899549044224207821149415964395... . - Vaclav Kotesovec, Jul 26 2014
a(n) = A304937(n) - A304937(n-1) for n>1, a(n) = 1 for n<2. - Gus Wiseman, May 22 2018

A242817 a(n) = B(n,n), where B(n,x) = Sum_{k=0..n} Stirling2(n,k)*x^k are the Bell polynomials (also known as exponential polynomials or Touchard polynomials).

Original entry on oeis.org

1, 1, 6, 57, 756, 12880, 268098, 6593839, 187104200, 6016681467, 216229931110, 8588688990640, 373625770888956, 17666550789597073, 902162954264563306, 49482106424507339565, 2901159958960121863952, 181069240855214001514460, 11985869691525854175222222

Offset: 0

Emanuele Munarini, May 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..370
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Cf. A035051, A292866, A346654, A346655.

Main diagonal of A189233 and of A292860.

A:= proc(n, k) option remember; `if`(n=0, 1, (1+
      add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, May 17 2016

Mathematica
```
Table[BellB[n, n], {n, 0, 100}]
```

a(n):=stirling2(n,0)+sum(stirling2(n,k)*n^k,k,1,n);
makelist(a(n),n,0,30);

a(n) = sum(k=0, n, stirling(n,k,2)*n^k); \\ Michel Marcus, Apr 20 2016

E.g.f.: x*f'(x)/f(x), where f(x) is the generating series for sequence A035051.
a(n) ~ (exp(1/LambertW(1)-2)/LambertW(1))^n * n^n / sqrt(1+LambertW(1)). - Vaclav Kotesovec, May 23 2014
Conjecture: It appears that the equation a(x)*e^x = Sum_{n=0..oo} ( (n^x*x^n)/n! ) is true for every positive integer x. - Nicolas Nagel, Apr 20 2016 [This is just the special case k=x of the formula  B(k,x) = e^(-x) * Sum_{n=0..oo} n^k*x^n/n!; see for example the World of Mathematics link. - Pontus von Brömssen, Dec 05 2020]
a(n) = n! * [x^n] exp(n*(exp(x)-1)). - Alois P. Heinz, May 17 2016
a(n) = [x^n] Sum_{k=0..n} n^k*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, May 31 2018

Name corrected by Pontus von Brömssen, Dec 05 2020

A007563 Number of rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

0, 1, 1, 3, 8, 25, 77, 258, 871, 3049, 10834, 39207, 143609, 532193, 1990163, 7503471, 28486071, 108809503, 417862340, 1612440612, 6248778642, 24309992576, 94905791606, 371691137827, 1459935388202, 5749666477454

Offset: 0

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 71, (3.4.13).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1600 (first 200 terms from T. D. Noe)
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 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]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 167
N. J. A. Sloane, Transforms

Cf. A007549, A030019, A035051, A035052, A035053.
Column k=2 of A144042.
Cf. A245566.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): c:= etr(b): a:= n-> if n=0 then 0 else c(n-1) fi: seq(a(n), n=0..25); # Alois P. Heinz, Sep 06 2008

etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a[0] = 0; a[n_] := etr[etr[a]][n-1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 28 2013, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); concat([0], v)} \\ Andrew Howroyd, May 20 2018

Shifts left when Euler transform is applied twice.

a(n) ~ c * d^n / n^(3/2), where d = 4.189610958393826965527036454524044275... (see A245566), c = 0.1977574301782950818433893126632477845870281049591883888... . - Vaclav Kotesovec, Jul 26 2014

New description from Christian G. Bower, Oct 15 1998

A007549 Number of increasing rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

1, 1, 3, 14, 89, 716, 6967, 79524, 1041541, 15393100, 253377811, 4596600004, 91112351537, 1959073928124, 45414287553455, 1129046241331316, 29965290866974493, 845605519848379436, 25282324544244718411, 798348403914242674980, 26549922456617388029641

Offset: 1

N. J. A. Sloane

In an increasing rooted graph, nodes are numbered and the numbers increase as you move away from the root.

(a(n+1)/a(n))/n tends to 1/A073003 = 1.676875... (same limit as A029768). - Vaclav Kotesovec, Jul 26 2014

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..410 (first 200 terms from Vincenzo Librandi)
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 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]

Cf. A007563, A030019, A035051-A035053.
Cf. A029768.
Row sums of A078341. Column k=1 of A264436.

exptr:= proc(p) local g; g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: end: b:= exptr(exptr(a)): a:= n-> `if`(n=0, 1, b(n-1)): seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008

exptr[p_] := Module[{g}, g[n_] := g[n] = p[n] + Sum[ Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; g]; b = exptr[ exptr[a] ]; a[n_] := If[n == 0, 1, b[n-1]]; Table[ a[n], {n, 1, 19}] (* Jean-François Alcover, May 10 2012, after Alois P. Heinz *)

Shifts left when exponentiated twice.
Conjecture: a(n) = Sum_{i=0..2^(n-2) - 1} b(i) for n > 1 with a(1) = 1 where b(n) = (L(n) + 2)*b(f(n)) + Sum_{k=0..L(n) - 1} (1 - R(n,k))*b(f(n) + 2^k*(1 - R(n,k))) for n > 0 with b(0) = 1, L(n) = A000523(n), f(n) = A053645(n) and where R(n,k) = floor(n/2^k) mod 2. Here R(n,k) is the (k+1)-th bit from the right side in the binary expansion of n. - Mikhail Kurkov, Jul 21 2024
Conjecture: a(n) = D^(n-1)(exp(x)) evaluated at x = 0, where D denotes the operator exp(x)*(1 + x)*d/dx. - Peter Bala, Feb 24 2025

New description from Christian G. Bower, Oct 15 1998

A035052 Number of sets of rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 134, 444, 1518, 5318, 18989, 68856, 252901, 938847, 3517082, 13278844, 50475876, 193014868, 741963015, 2865552848, 11113696421, 43266626430, 169019868095, 662337418989, 2602923589451, 10256100717875

Offset: 0

Christian G. Bower, Oct 15 1998

nonn

T. D. Noe, Table of n, a(n) for n=0..200
Loic Foissy, The Hopf algebra of Fliess operators and its dual pre-Lie algebra, 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 862
N. J. A. Sloane, Transforms

Cf. A007549, A007563, A030019, A035051, A035053.

Cf. A245566.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: b:= etr(aa): c:= etr(b): aa:= n-> if n=0 then 0 else c(n-1) fi: a:= etr(aa): seq(a(n), n=0..25); # Alois P. Heinz, Sep 09 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b = etr[aa]; c = etr[b]; aa = Function[{n}, If[n == 0, 0, c[n-1]]]; a = etr[aa]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); concat([1], EulerT(v))} \\ Andrew Howroyd, May 20 2018

Euler transform of A007563.

a(n) ~ c * d^n / n^(3/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.35683683547585... . - Vaclav Kotesovec, Jul 26 2014

A210586 Triangle T(n,k) read by rows: T(n,k) is the number of rooted hypertrees on n labeled vertices with k hyperedges, n >= 2, k >= 1.

Original entry on oeis.org

2, 3, 9, 4, 48, 64, 5, 175, 750, 625, 6, 540, 5400, 12960, 7776, 7, 1519, 30870, 156065, 252105, 117649, 8, 4032, 154112, 1433600, 4587520, 5505024, 2097152, 9, 10287, 704214, 11160261, 62001450, 141363306, 133923132, 43046721, 10, 25500, 3025000, 77700000, 695100000, 2646000000, 4620000000, 3600000000, 1000000000

Offset: 2

Peter Bala, Mar 26 2012

A hypergraph H is a pair (V,E) consisting of a finite set V of vertices and a set E of hyperedges given by subsets of V containing at least two elements. A walk in a hypergraph H connecting vertices v0 and vn is a sequence v0, e1, v1, e2, ... , v(n-1), en, vn, where each vi is in V and each ei is in E and for each ei the set {v(i-1),vi} is contained in ei. If for every pair of vertices v and v0 there is a walk in H starting at v and ending at v0 then H is called connected. A walk is a cycle if it contains at least two edges, all of the ei are distinct and all of the vi are distinct except v0 = vn. A connected hypergraph with no cycles is called a hypertree. A rooted hypertree is a hypertree in which one particular vertex is selected as being the root. For the enumeration of unrooted hypertrees see A210587.

			Triangle begins
.n\k.|....1.....2......3.......4.......5.......6
= = = = = = = = = = = = = = = = = = = = = = = = =
..2..|....2
..3..|....3.....9
..4..|....4....48.....64
..5..|....5...175....750.....625
..6..|....6...540...5400...12960....7776
..7..|....7..1519..30870..156065..252105..117649
...
Example of a hypertree with two hyperedges, one a 2-edge {3,4} and one a 3-edge {1,2,3}.
........__________........................
......./..........\.______................
......|....1...../.\......\...............
......|.........|.3.|....4.|..............
......|....2.....\./______/...............
.......\__________/.......................
..........................................
T(4,2) = 48. The twelve unrooted hypertrees on 4 vertices {1,2,3,4} with 2 hyperedges (one a 2-edge and one a 3-edge) have hyperedges:
{1,2,3} and {3,4}; {1,2,3} and {2,4}; {1,2,3} and {1,4};
{1,2,4} and {1,3}; {1,2,4} and {2,3}; {1,2,4} and {3,4};
{1,3,4} and {1,2}; {1,3,4} and {2,3}; {1,3,4} and {2,4};
{2,3,4} and {1,2}; {2,3,4} and {1,3}; {2,3,4} and {1,4}.
Choosing one of the four vertices as the root gives a total of 4x12 = 48 rooted hypertrees on 4 vertices.

Andrew Howroyd, Table of n, a(n) for n = 2..1276
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.
J. McCammond and J. Meier, The hypertree poset and the l^2-Betti numbers of the motion group of the trivial link, Mathematische Annalen 328 (2004), no. 4, 633-652.

Cf. A035051 (row sums). Cf. A210587, A048993.

with(combinat):
A210586 := (n, k) -> n^k*stirling2(n-1, k):
for n from 2 to 10 do seq(A210586(n, k), k = 1..n-1) end do;
# Peter Bala, Oct 28 2015

T(n,k) = {n^k*stirling(n-1,k,2)}
for(n=2, 10, for(k=1, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Aug 28 2018

T(n,k) = n^k*Stirling2(n-1,k). T(n,k) = n*A210587(n,k).
E.g.f. A(x,t) = t + 2*x*t^2/2! + (3*x + 9*x^2)*t^3/3! + ... satisfies A(x,t) = t*exp(x*(exp(A(x,t)) - 1)).
Dobinski-type formula for the row polynomials: R(n,x) = exp(-n*x)*Sum_{k = 0..inf} n^k*k^(n-1)x^k/k!.
Row sums A035051.
The e.g.f. is essentially the series reversion of t/F(x,t) w.r.t. t, where F(x,t) = exp(x*(exp(t) - 1)) is the e.g.f. of the Stirling numbers of the second kind A048993. - Peter Bala, Oct 28 2015

cp's OEIS Frontend

A030438 a(n) = A030019(n)*n! (or A035051*(n-1)!).

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A030019 Number of labeled spanning trees in the complete hypergraph on n vertices (all hyperedges having cardinality 2 or greater).

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Mathematica

PARI

PARI

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A035053 Number of connected graphs on n unlabeled nodes where every block is a complete graph.

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Maple

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PARI

Formula

A242817 a(n) = B(n,n), where B(n,x) = Sum_{k=0..n} Stirling2(n,k)*x^k are the Bell polynomials (also known as exponential polynomials or Touchard polynomials).

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Maple

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Maxima

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A007563 Number of rooted connected graphs where every block is a complete graph.

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Maple

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A007549 Number of increasing rooted connected graphs where every block is a complete graph.

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Maple

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A035052 Number of sets of rooted connected graphs where every block is a complete graph.

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A030438 a(n) = A030019(n)n! (or A035051(n-1)!).