A007563 -id:A007563 - cp's OEIS frontend

A245566 Decimal expansion of a constant related to A007563.

4, 1, 8, 9, 6, 1, 0, 9, 5, 8, 3, 9, 3, 8, 2, 6, 9, 6, 5, 5, 2, 7, 0, 3, 6, 4, 5, 4, 5, 2, 4, 0, 4, 4, 2, 7, 5, 9, 4, 2, 3, 8, 9, 9, 2, 5, 9, 1, 5, 9, 3, 6, 5, 9, 4, 1, 3, 2, 8, 5, 7, 7, 4, 2, 5, 9, 8, 9, 8, 7, 0, 6, 4, 9, 1, 2, 0, 6, 1, 9, 9, 0, 1, 7, 6, 0, 7, 4, 0, 6, 3, 9, 5, 8, 9, 6, 8, 5, 6, 3, 3, 8, 2, 5, 3

Offset: 1

Vaclav Kotesovec, Jul 26 2014

			4.18961095839382696552703645452404427594238992591593659413285774...

Cf. A007563, A035052, A035053, A134955.

Equals lim n -> infinity A007563(n)^(1/n).
Equals lim n -> infinity A035052(n)^(1/n).
Equals lim n -> infinity A035053(n)^(1/n).
Equals lim n -> infinity A134955(n)^(1/n).

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

A134955 Number of "hyperforests" on n unlabeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 50, 128, 351, 1009, 3035, 9464, 30479, 100712, 340072, 1169296, 4082243, 14438577, 51643698, 186530851, 679530937, 2494433346, 9219028889, 34280914106, 128179985474, 481694091291, 1818516190252, 6894350122452

Offset: 0

Don Knuth, Jan 26 2008

nonn

A hyperforest is an antichain of finite nonempty sets (edges) whose connected components are hypertrees. It is spanning if all vertices are covered by some edge. However, it is common to represent uncovered vertices as singleton edges. For example, {{1,2},{1,4}} and {{3},{1,2},{1,4}} may represent the same hyperforest, the former being free of singletons (see example 2) and the latter being spanning (see example 1). This is different from a hyperforest with singleton edges allowed, which is one whose non-singleton edges only are required to form an antichain. For example, {{1},{2},{1,3},{2,3}} is a hyperforest with singleton edges allowed. - Gus Wiseman, May 22 2018

Equivalently, number of block graphs on n nodes, that is, graphs where every block is a complete graph. These graphs 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(4) = 9 spanning hyperforests are the following:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{1,2},{3,4}}
  {{1,4},{2,3,4}}
  {{1},{2},{3,4}}
  {{1},{2,4},{3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1},{2},{3},{4}}
Non-isomorphic representatives of the a(4) = 9 hyperforests spanning up to 4 vertices without singleton edges are the following:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,2,3,4}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}
  {{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
(End)

D. E. Knuth: The Art of Computer Programming, Volume 4, Generating All Combinations and Partitions Fascicle 3, Section 7.2.1.4. Generating all partitions. Page 38, Algorithm H. - Washington Bomfim, Sep 25 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
N. J. A. Sloane, Transforms
Wikipedia, Block graph

Cf. A030019, A035053 (hypertrees), A048143, A054921, A134954 (labeled case), A134955, A134957, A144959, A245566, A304716, A304717, A304867, A304911.

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): aa:= proc(n) option remember; B(n)+C(n) -add(B(k)*C(n-k), k=0..n) end: a:= etr(aa): seq(a(n), n=0..27); # 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[B]; c = etr[b]; B[n_] := If[n == 0, 0, c[n-1]]; CC = etr[B]; aa[n_] := aa[n] = B[n]+CC[n]-Sum[B[k]*CC[n-k], {k, 0, n}]; a = etr[aa]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz*)

\\ here b 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)); concat([1], EulerT(Vec(x*Ser(EulerT(u))*(1-x*Ser(u)))))} \\ Andrew Howroyd, May 22 2018

Euler transform of A035053. - N. J. A. Sloane, Jan 30 2008
a(n) = Sum of prod_{k=1}^n\,{A035053(k) + c_k -1 /choose c_k} over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0. - Washington Bomfim, Sep 25 2008
a(n) ~ c * d^n / n^(5/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.36483930544... . - Vaclav Kotesovec, Jul 26 2014

A144959 A134955(n) - A134955(n-1). Number of hyperforests spanning n unlabeled nodes without isolated vertices.

Original entry on oeis.org

1, 0, 1, 2, 5, 11, 30, 78, 223, 658, 2026, 6429, 21015, 70233, 239360, 829224, 2912947, 10356334, 37205121, 134887153, 493000086, 1814902409, 6724595543, 25061885217, 93899071368, 353514105817, 1336822098961, 5075833932200

Offset: 0

Washington Bomfim, Sep 27 2008

nonn

a(n) is the number of hyperforests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A134955(n-1) counts the hyperforests of order n with one or more isolated nodes.

			From _Gus Wiseman_, May 21 2018: (Start)
Non-isomorphic representatives of the a(5) = 11 hyperforests are the following:
  {{1,2,3,4,5}}
  {{1,2},{3,4,5}}
  {{1,5},{2,3,4,5}}
  {{1,2,5},{3,4,5}}
  {{1,2},{2,5},{3,4,5}}
  {{1,2},{3,5},{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)

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144958 (unlabeled forests without isolated vertices), A144959, A304716, A304717, A304867, A304911.

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];
c[1] = 0; c[n_] := b[n] + etr[b][n] - Sum[b[k]*etr[b][n-k], {k, 0, n}];
a = etr[c];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jul 12 2018, after Alois P. Heinz's code for A035053 *)

\\ here b 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)); concat([1], EulerT(concat([0], Vec(Ser(EulerT(u))*(1-x*Ser(u))-1))))} \\ Andrew Howroyd, May 22 2018

Euler transform of b(1) = 0, b(n > 1) = A035053(n). - Gus Wiseman, May 21 2018

A144042 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Euler transform applied k times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 9, 1, 1, 5, 13, 25, 20, 1, 1, 6, 19, 51, 77, 48, 1, 1, 7, 26, 89, 197, 258, 115, 1, 1, 8, 34, 141, 410, 828, 871, 286, 1, 1, 9, 43, 209, 751, 2052, 3526, 3049, 719, 1, 1, 10, 53, 295, 1260, 4337, 10440, 15538, 10834, 1842, 1, 1, 11, 64

Offset: 1

Alois P. Heinz, Sep 08 2008

			Square array begins:
    1,   1,    1,     1,     1,     1,      1,      1, ...
    1,   1,    1,     1,     1,     1,      1,      1, ...
    2,   3,    4,     5,     6,     7,      8,      9, ...
    4,   8,   13,    19,    26,    34,     43,     53, ...
    9,  25,   51,    89,   141,   209,    295,    401, ...
   20,  77,  197,   410,   751,  1260,   1982,   2967, ...
   48, 258,  828,  2052,  4337,  8219,  14379,  23659, ...
  115, 871, 3526, 10440, 25512, 54677, 106464, 192615, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
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]
N. J. A. Sloane, Transforms

Columns k=1-10 give: A000081, A007563, A144035, A144036, A144037, A144038, A144039, A144040, A144041, A305727.
Rows n=2-4 give: A000012, A000027, A034856.
Main diagonal gives A305725.
Cf. A316101.

etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
        add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
      end end:
g:= proc(k) option remember; local b, t; b[0]:= j->
      `if`(j<2, j, b[k](j-1)); for t to k do
       b[t]:= etr(b[t-1]) od: eval(b[0])
    end:
A:= (n, k)-> g(k)(n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # revised Alois P. Heinz, Aug 27 2018

etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[Table[A[n, 1 + d-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

A035051 Number of labeled rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

0, 1, 2, 12, 116, 1555, 26682, 558215, 13781448, 392209380, 12641850510, 455198725025, 18109373455164, 788854833679549, 37343190699472322, 1908871649888004240, 104789417805394595600, 6148562290130009617619

Offset: 0

Christian G. Bower, Oct 15 1998

Equivalently, rooted labeled spanning trees in the complete hypergraph on n vertices (all hyperedges having cardinality 2 or greater).

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

T. D. Noe, Table of n, a(n) for n=0..100
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, 2016.
I. M. Gessel and L. H. Kalikow, Hypergraphs and a functional equation...
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 864
D. M. Warme, Spanning Trees in Hypergraphs with Applications to Steiner Trees. PhD thesis, University of Virginia, 1998.

Cf. A007549, A007563, A030019, A038052, A038053, A030438. A052888, A210586.

f[n_] := Sum[ n^i*StirlingS2[n - 1, i], {i, 0, n - 1}]; Array[f, 18, 0] (* Robert G. Wilson v, Apr 05 2012 *)
Table[If[n == 0, 0, BellB[n - 1, n]], {n, 0, 100}] (* Emanuele Munarini, May 23 2014 *)

a(n):=if n=0 then 0 else sum(stirling2(n-1,k)*n^k,k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, May 23 2014 */

for(n=0,30, print1(sum(k=0,n-1, stirling(n-1,k,2)*n^k), ", ")) \\ G. C. Greubel, Nov 17 2017

Recurrence: a(1) = 1, a(n) = Sum_{k=1}^{n-1} Bell(k) / k! Sum_{a_j > 0, Sum_{j=1}^k a_j = n-1} {{n-1} choose {a_1, a_2, ..., a_k }} \prod_{j=1}^k a(a_j) for n > 1, where Bell(k) = A000110(k). - Warren D. Smith, Feb 23 1998
a(n) = Sum_{i=0...n-1} S(n-1, i) n^i, where S(N, M) are Stirling numbers of the second kind - David Warme, Mar 25 1998
E.g.f. satisfies A(x)=x*exp(exp(A(x))-1).
Let X_{mu} be a Poisson random variable with mean mu: P(X_{mu} = K) = e^{-mu} mu^K / K!. The n-th moment of X_{mu} is E[X_{mu}^n] = sum_{i=0}^n S(n, i) mu^i. Therefore a(n) = E[X_n^{n-1}]. - Langworth Withers, May 25 2000
Dobinski-type formula: a(n) = 1/e^n*sum {k = 0..inf} n^k*k^(n-1)/k!. Cf. A030019 and A052888. For a refinement of this sequence see A210586. - Peter Bala, Apr 05 2012
a(n) ~ exp((1/LambertW(1)-2)*n) * n^(n-1) / (sqrt(1+LambertW(1)) * LambertW(1)^(n-1)). - Vaclav Kotesovec, Jan 22 2014

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

A007561 Number of asymmetric rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

0, 1, 1, 1, 3, 6, 16, 43, 120, 339, 985, 2892, 8606, 25850, 78347, 239161, 734922, 2271085, 7054235, 22010418, 68958139, 216842102, 684164551, 2165240365, 6871792256, 21865189969, 69737972975, 222915760126, 714001019626, 2291298553660, 7366035776888

Offset: 0

N. J. A. Sloane

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

Alois P. Heinz, Table of n, a(n) for n = 0..1900
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]
N. J. A. Sloane, Transforms

Cf. A007563, A035079-A035081.

Column k=2 of A316101.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*g(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 0, b(n-1, n-1)):
seq(a(n), n=0..40); # Alois P. Heinz, May 19 2013

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n<1, 0, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 0, 40}] (* Jean-François Alcover, Feb 11 2014, after Alois P. Heinz *)

Shifts left when weigh-transform applied twice.

a(n) ~ c * d^n / n^(3/2), where d = 3.382016466020272807429818743..., c = 0.161800727760188847021075748... . - Vaclav Kotesovec, Jul 26 2014

Additional comments from Christian G. Bower

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

A318494 Number of rooted simple connected graphs on n unlabeled nodes where every block is a complete graph with nonroot nodes of two colors.

Original entry on oeis.org

1, 2, 10, 50, 285, 1696, 10647, 68842, 456922, 3091546, 21252396, 147992264, 1041779912, 7401119718, 52996414666, 382095695324, 2771458821772, 20209364313202, 148064910503435, 1089415620952020, 8046283404651000, 59635009544475814, 443380411766040664

Offset: 1

Andrew Howroyd, Aug 27 2018

nonn

Number of rooted spanning hypertrees on n unlabeled nodes with edges of size 1 allowed.

Shifts left when Euler transform is applied twice to double this sequence.

			a(3) = 10 because there are three possible rooted graphs which are illustrated below and these can be colored up to isomorphism in 3, 3 and 4 ways respectively.
  o---o   o   o   o---o
   \ /     \ /     \
    *       *       *

Alois P. Heinz, Table of n, a(n) for n = 1..1116 (first 200 terms from Andrew Howroyd)

Cf. A007563, A134959.

b:= ((proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
        add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
      end end)@@2)(2*a):
a:= n-> b(n-1):
seq(a(n), n=1..25);  # Alois P. Heinz, Aug 27 2018

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[n_] := b[n-1];
b = etr@etr@(2a[#]&);
Array[a, 25] (* Jean-François Alcover, Nov 01 2020 *)

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(2*v)))); v}

cp's OEIS Frontend

A245566 Decimal expansion of a constant related to A007563.

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

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A134955 Number of "hyperforests" on n unlabeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

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A144959 A134955(n) - A134955(n-1). Number of hyperforests spanning n unlabeled nodes without isolated vertices.

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A144042 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Euler transform applied k times.

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A035051 Number of labeled rooted connected graphs where every block is a complete graph.

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Maxima

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

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