A052888 -id:A052888 - cp's OEIS frontend

A321227 Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n.

0, 1, 3, 6, 17, 43, 147, 458, 1729, 6445, 27011

Offset: 0

Gus Wiseman, Oct 31 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.

			The a(1) = 1 through a(4) = 17 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
         {{1},{1}}  {{1,2,3}}      {{1,1,2,2}}
                    {{1},{1,1}}    {{1,1,2,3}}
                    {{1},{1,2}}    {{1,2,3,4}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1,1},{1,1}}
                                   {{1},{1,1,2}}
                                   {{1,1},{1,2}}
                                   {{1},{1,2,2}}
                                   {{1},{1,2,3}}
                                   {{1,2},{1,3}}
                                   {{2},{1,1,2}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{1,2}}
                                   {{1},{2},{1,2}}
                                   {{1},{1},{1},{1}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321228, A321229, A321231.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
mensity[c_]:=Total[(Length[Union[#]]-1&)/@c]-Length[Union@@c];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Sum[Length[Select[mps[m],And[mensity[#]==-1,Length[csm[#]]==1]&]],{m,strnorm[n]}],{n,0,8}]

A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

Original entry on oeis.org

1, 3, 1, 16, 1, 125, 15, 1, 1296, 1, 16807, 735, 140, 1, 262144, 1, 4782969, 76545, 1890, 1, 100000000, 112000, 1, 2357947691, 13835745, 33264, 1, 61917364224, 1, 1792160394037, 3859590735, 270670400, 35135100, 720720, 1, 56693912375296, 1, 1946195068359375

Offset: 1

Gus Wiseman, Jul 03 2019

A hypertree is a connected hypergraph of density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices. A hypergraph is k-uniform if its edges all have size k. The span of a hypertree is the union of its edges.

			Triangle begins:
           1
           3          1
          16          1
         125         15          1
        1296          1
       16807        735        140          1
      262144          1
     4782969      76545       1890          1
   100000000     112000          1
  2357947691   13835745      33264          1
The T(4,2) = 15 hypertrees:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}

Alois P. Heinz, Rows n = 1..185, flattened

Row sums are A320444.
Column d = 1 is A000272.
Cf. A030019, A027750, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{n,10},{d,Divisors[n]}]

T(n, k) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1), where d = A027750(n, k).

Edited by Peter Munn, Mar 05 2025

A334986 a(n) = exp(n) * Sum_{k>=0} (-1)^k * n^(k-1) * k^(n-1) / k!.

Original entry on oeis.org

1, -1, 2, -5, 9, 53, -1107, 12983, -116470, 560049, 8370713, -346902877, 7551856337, -117404648467, 913399734614, 22560135521007, -1393700803877939, 44331044030953865, -979905458659247779, 10462396536804802459, 367799071887303276422, -30046998012662824941947

Offset: 1

Ilya Gutkovskiy, May 18 2020

sign

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A030019, A052888, A292866.

Table[Sum[(-1)^k StirlingS2[n - 1, k] n^(k - 1), {k, 0, n - 1}], {n, 1, 22}]
Table[BellB[n - 1, -n]/n, {n, 1, 22}]

a(n)={sum(k=0, n-1, (-1)^k * stirling(n-1,k,2) * n^(k-1))} \\ Andrew Howroyd, May 18 2020

a(n) = Sum_{k=0..n-1} (-1)^k * Stirling2(n-1,k) * n^(k-1).

a(n) = BellPolynomial_(n-1)(-n) / n.

A322112 Number of non-isomorphic self-dual connected multiset partitions of weight n with no singletons and multiset density -1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 4, 4, 9, 9

Offset: 0

Gus Wiseman, Nov 26 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. A multiset partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(10) = 9 multiset partitions:
  {{11}}  {{111}}  {{1111}}  {{11111}}    {{111111}}    {{1111111}}
                             {{11}{122}}  {{22}{1122}}  {{111}{1222}}
                                                        {{22}{11222}}
                                                        {{11}{12}{233}}
.
  {{11111111}}      {{111111111}}        {{1111111111}}
  {{111}{11222}}    {{1111}{12222}}      {{1111}{112222}}
  {{22}{112222}}    {{22}{1122222}}      {{22}{11222222}}
  {{11}{122}{233}}  {{222}{111222}}      {{222}{1112222}}
                    {{11}{11}{12233}}    {{111}{122}{2333}}
                    {{11}{113}{2233}}    {{22}{113}{23333}}
                    {{12}{111}{2333}}    {{22}{1133}{2233}}
                    {{22}{113}{2333}}    {{33}{33}{112233}}
                    {{12}{13}{22}{344}}  {{11}{14}{223}{344}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A321155, A321255, A322111.

A357394 E.g.f. satisfies A(x) = exp(x * exp(2 * A(x))) - 1.

Original entry on oeis.org

0, 1, 5, 55, 953, 22651, 685525, 25222359, 1093148145, 54549313651, 3080446982221, 194213549023407, 13522789698386281, 1030619149263349387, 85336828127587240261, 7628421633465044832391, 732208108150442899232737, 75108533335473988089786147

Offset: 0

Seiichi Manyama, Sep 26 2022

nonn

Index entries for reversions of series

Cf. A052888, A357395.

Cf. A357335.

Table[Sum[(2*n)^(k-1) * StirlingS2[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2022 *)

a(n) = sum(k=1, n, (2*n)^(k-1)*stirling(n, k, 2));

a(n) = Sum_{k=1..n} (2 * n)^(k-1) * Stirling2(n,k).
a(n) ~ n^(n-1) / (2 * sqrt(1 + LambertW(1/2)) * LambertW(1/2)^n * exp(n*(3 - 1/LambertW(1/2)))). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( exp(-2*x) * log(1 + x) ). - Seiichi Manyama, Sep 10 2024

A357395 E.g.f. satisfies A(x) = exp(x * exp(3 * A(x))) - 1.

Original entry on oeis.org

0, 1, 7, 109, 2677, 90226, 3873007, 202134997, 12427851625, 879806921041, 70486590597331, 6304879010400202, 622838214328334077, 67347956304168803173, 7911963620634266270071, 1003477119181096373029261, 136658009168055564212000209, 19889317400287888238121299854

Offset: 0

Seiichi Manyama, Sep 26 2022

nonn

Index entries for reversions of series

Cf. A052888, A357394.

Cf. A357336.

Table[Sum[(3*n)^(k-1) * StirlingS2[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2022 *)

a(n) = sum(k=1, n, (3*n)^(k-1)*stirling(n, k, 2));

a(n) = Sum_{k=1..n} (3 * n)^(k-1) * Stirling2(n,k).
a(n) ~ n^(n-1) / (3 * sqrt(1 + LambertW(1/3)) * LambertW(1/3)^n * exp(n*(4 - 1/LambertW(1/3)))). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( exp(-3*x) * log(1 + x) ). - Seiichi Manyama, Sep 10 2024

A367752 Number of shapes of labeled rooted hypertrees with n vertices.

Original entry on oeis.org

1, 1, 4, 29, 256, 3007, 42932, 721121, 13982563, 306967231, 7527903208, 203977383469, 6051630040496, 195111205542541, 6792697846367791, 253966747582533681, 10149075292428481965, 431705938073882999275, 19474660918369182445456, 928660364396786865580881

Offset: 1

Paul Laubie, Nov 29 2023

nonn

The shape of a labeled rooted hypertree is a labeled rooted hypertrees where we replace all the maximal subtrees by a corolla rooted on a new unlabeled black vertex.
If we remove the black vertices that are the parent of only 1 white vertex, we obtain labeled rooted hypertrees with black and white vertices such that:
  - black vertices are unlabeled;
  - black vertices have at least two children;
  - the children of a black vertex are white, and are connected to it via simple edges (edges connecting only two vertices);
  - the children of a white vertex are connected to it via hyperedges (edges connecting strictly more than two vertices).

			For n = 3 the a(3) = 4 solutions are:
  - the corolla with a black root which have 3 white children,
  - and the 3 possible labeling of the hypertree with a white root which have 2 white children connected to it via a hyperedge.

Cf. A052888, A210586, A364709, A364816.

my(x='x+O('x^30)); Vec(serlaplace(serreverse(log(1+x)*exp(-exp(x)+x+1)))) \\ Michel Marcus, Nov 30 2023

R.=PowerSeriesRing(QQ);(ln(1+t)*exp(-exp(t)+t+1)).reverse().egf_to_ogf().list()[1:]

E.g.f.: series reversion of log(1+x)*exp(-exp(x)+x+1).

A370949 Triangle read by rows: T(n,k) is the number of forests of labeled rooted Greg hypertrees with n white vertices and k black vertices, 0 <= k < n.

Original entry on oeis.org

1, 3, 1, 19, 16, 3, 189, 268, 115, 15, 2576, 5221, 3655, 1050, 105, 44683, 118599, 117236, 54040, 11655, 945, 941977, 3102184, 3996384, 2581138, 883575, 152460, 10395, 23388025, 92149019, 147043422, 123318510, 58806055, 15980580, 2297295, 135135

Offset: 1

Paul Laubie, Mar 06 2024

A rooted Greg hypertree is a hypertree with black and white vertices such that white vertices are labeled, black vertices are unlabeled, and each black vertex has at least two children.

See A048160 for the analog sequence for Greg trees.

			Triangle T(n,k) begins:
n\k    0     1     2     3     4 ...
1      1;
2      3,    1;
3     19,   16,    3;
4    189,  268,  115,   15;
5   2576, 5221, 3655, 1050,  105;
...

Paul Laubie, Hypertrees and embedding of the FMan operad, arXiv:2401.17439 [math.QA], 2024.

Cf. A048160, A052888 (k=0), A001147 (k=n-1).

Row sums are A364816.

T(n)={my(x='x+O('x^(n+1))); [Vecrev(p) | p <- Vec(serlaplace(serreverse( (log(1+x) - y*exp(x) + y*x + y)*exp(-x) )))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Mar 06 2024

E.g.f.: series reversion in t of (log(1+t) - u*exp(t) + u*t + u)*exp(-t), where the formal variable u encodes the number of black vertices.
T(n,0) = A052888(n).
T(n,n-1) = A001147(n).

A224079 E.g.f. is series reversion of log(1+x)/cosh(x).

Original entry on oeis.org

1, 1, 4, 25, 191, 1981, 24515, 357393, 6014944, 114374701, 2429126965, 56973837097, 1462548099325, 40790689845725, 1228180553509096, 39706476998683809, 1371869867621426343, 50445615936195883981, 1967026296214873286071, 81070802180747506986681

Offset: 1

Paul D. Hanna, Jul 20 2013

nonn

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 191*x^5/5! + 1981*x^6/6! +...

Cf. A052888.

Rest[CoefficientList[InverseSeries[Series[Log[1+x]/Cosh[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

{a(n)=local(X=x+x*O(x^n));n!*polcoeff(serreverse(log(1+X)/cosh(X)), n)}
for(n=1,25,print1(a(n),", "))

E.g.f. satisfies: 1 + A(x) = exp(x*cosh(A(x))).

a(n) ~ n^(n-1) * ((1+s)*sinh(s))^n * sqrt((1+s)/(1+s+tanh(s))) / exp(n), where s = 0.96996536567590308324... is the root of the equation (1+s)*log(1+s)*tanh(s) = 1. - Vaclav Kotesovec, Jan 13 2014

A332915 Decimal expansion of the constant W(1) + 1/W(1), where W is Lambert's function.

Original entry on oeis.org

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

Offset: 1

Martin Renner, Mar 02 2020

The graph of the exponential function exp(x) moved to the right by W(1) + 1/W(1) touches the graph of the natural logarithm log(x) at point (x,y) = (1/W(1), W(1)) = (A030797, A030178).

			2.33036612476168058322517043916206263018983377385398...

Cf. A030178, A030797, A052888.

evalf[200](LambertW(1) + 1/LambertW(1));

RealDigits[N[LambertW[1] + 1/LambertW[1], 120]][[1]] (* Vaclav Kotesovec, Mar 02 2020 *)

my(x=lambertw(1)); x+1/x \\ Michel Marcus, Mar 02 2020

Equals A030178 + A030797 = A030178 + 1/A030178 = 1/A030797 + A030797.

Equals 2 + Integral_{x=0..1} W(x) dx. - Amiram Eldar, Jul 18 2021

cp's OEIS Frontend

A321227 Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A334986 a(n) = exp(n) * Sum_{k>=0} (-1)^k * n^(k-1) * k^(n-1) / k!.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A322112 Number of non-isomorphic self-dual connected multiset partitions of weight n with no singletons and multiset density -1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A357394 E.g.f. satisfies A(x) = exp(x * exp(2 * A(x))) - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A357395 E.g.f. satisfies A(x) = exp(x * exp(3 * A(x))) - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A367752 Number of shapes of labeled rooted hypertrees with n vertices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

SageMath

Formula

A370949 Triangle read by rows: T(n,k) is the number of forests of labeled rooted Greg hypertrees with n white vertices and k black vertices, 0 <= k < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs