A052888 -id:A052888 - cp's OEIS frontend

A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

0, 1, 2, 5, 12, 28, 78, 202, 578, 1650, 4904

Offset: 0

Gus Wiseman, Nov 01 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 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(1) = 1 through a(5) = 28 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}      {{1,1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}      {{1,1,2,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}      {{1,2,2,2,2}}
                  {{1},{1,1}}  {{1,2,3,3}}      {{1,2,2,3,3}}
                  {{2},{1,2}}  {{1,2,3,4}}      {{1,2,3,3,3}}
                               {{1},{1,1,1}}    {{1,2,3,4,4}}
                               {{1},{1,2,2}}    {{1,2,3,4,5}}
                               {{1,2},{2,2}}    {{1},{1,1,1,1}}
                               {{1,3},{2,3}}    {{1,1},{1,1,1}}
                               {{2},{1,2,2}}    {{1,1},{1,2,2}}
                               {{3},{1,2,3}}    {{1},{1,2,2,2}}
                               {{1},{2},{1,2}}  {{1,2},{2,2,2}}
                                                {{1,2},{2,3,3}}
                                                {{1,3},{2,3,3}}
                                                {{1,4},{2,3,4}}
                                                {{2},{1,1,2,2}}
                                                {{2},{1,2,2,2}}
                                                {{2},{1,2,3,3}}
                                                {{2,2},{1,2,2}}
                                                {{3},{1,2,3,3}}
                                                {{3,3},{1,2,3}}
                                                {{4},{1,2,3,4}}
                                                {{1},{1,2},{2,2}}
                                                {{1},{2},{1,2,2}}
                                                {{2},{1,2},{2,2}}
                                                {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321194, A321228, A321229, A321254.

A322111 Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 13, 13, 37, 37

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(1) = 1 through a(8) = 13 multiset partitions:
  {{1}}                    {{1,1}}
.
  {{1,1,1}}                {{1,1,1,1}}
  {{2},{1,2}}              {{2},{1,2,2}}
.
  {{1,1,1,1,1}}            {{1,1,1,1,1,1}}
  {{1,1},{1,2,2}}          {{2},{1,2,2,2,2}}
  {{2},{1,2,2,2}}          {{2,2},{1,1,2,2}}
  {{2},{1,3},{2,3}}        {{2},{1,3},{2,3,3}}
  {{3},{3},{1,2,3}}        {{3},{3},{1,2,3,3}}
.
  {{1,1,1,1,1,1,1}}        {{1,1,1,1,1,1,1,1}}
  {{1,1,1},{1,2,2,2}}      {{1,1,1},{1,1,2,2,2}}
  {{2},{1,2,2,2,2,2}}      {{2},{1,2,2,2,2,2,2}}
  {{2,2},{1,1,2,2,2}}      {{2,2},{1,1,2,2,2,2}}
  {{1,1},{1,2},{2,3,3}}    {{1,1},{1,2,2},{2,3,3}}
  {{2},{1,3},{2,3,3,3}}    {{2},{1,3},{2,3,3,3,3}}
  {{2},{2,2},{1,2,3,3}}    {{2},{1,3,3},{2,2,3,3}}
  {{3},{1,2,2},{2,3,3}}    {{3},{3},{1,2,3,3,3,3}}
  {{3},{3},{1,2,3,3,3}}    {{3},{3,3},{1,2,2,3,3}}
  {{1},{1},{1,4},{2,3,4}}  {{2},{1,3},{2,4},{3,4,4}}
  {{2},{1,3},{2,4},{3,4}}  {{3},{3},{1,2,4},{3,4,4}}
  {{3},{4},{1,4},{2,3,4}}  {{3},{4},{1,4},{2,3,4,4}}
  {{4},{4},{4},{1,2,3,4}}  {{4},{4},{4},{1,2,3,4,4}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A318697, A319616, A321155, A321255.

A357347 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(2 * A(x)).

Original entry on oeis.org

0, 1, 7, 103, 2385, 75756, 3064239, 150689953, 8729691693, 582299930167, 43956280309659, 3704637865439380, 344825037782332457, 35131983926187957173, 3888817094785288023367, 464724955485177444101895, 59631976064836824117227621, 8177487264101392841050876136

Offset: 0

Seiichi Manyama, Sep 25 2022

nonn

Index entries for reversions of series

Cf. A052888, A357346, A357348, A357424.

Cf. A355762.

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

a(n) = Sum_{k=1..n} (n+2*k)^(k-1) * Stirling2(n,k).

E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(-2*x)) ). - Seiichi Manyama, Sep 09 2024

A364709 Triangle read by rows: T(n,k) is the number of forests of labeled rooted hypertrees with n vertices and weight k, 0 <= k < n.

Original entry on oeis.org

1, 2, 1, 9, 9, 1, 64, 96, 28, 1, 625, 1250, 625, 75, 1, 7776, 19440, 14040, 3240, 186, 1, 117649, 352947, 336140, 120050, 14749, 441, 1, 2097152, 7340032, 8716288, 4300800, 870912, 61824, 1016, 1, 43046721, 172186884, 245525742, 156243654, 45605511, 5664330, 245025, 2295, 1

Offset: 1

Paul Laubie, Oct 20 2023

The weight is the number of hypertrees minus 1 plus the weight of each hyperedge which is the number of vertices it connects minus 2.

T(n,k) is also the dimension of the operad ComPreLie in arity n with k commutative products.

			Triangle T(n,k) begins:
n\k   0     1    2    3    4 ...
1     1;
2     2,    1;
3     9,    9,   1;
4    64,   96,  28,   1;
5   625, 1250, 625,  75,   1;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Cf. A000169 (k=0), A081131 (k=1).
Row sums are A052888.
Series reversion as e.g.f of A111492 with an offset of 1.

T(n) = my(x='x+O('x^(n+1))); [Vecrev(p) | p<-Vec(serlaplace( serreverse(log(1+x*y)*exp(-x)/y )))]
{my(A=T(10)); for(n=1, #A, print(A[n]))} \\ Andrew Howroyd, Oct 20 2023

E.g.f: series reversion in t of (log(1+x*t)/x)*exp(-t).
T(n,0) = n^(n-1).
T(n,n-1) = 1.

a(23) corrected by Andrew Howroyd, Jan 01 2024

A364816 Number of labeled forests of rooted Greg hypertrees with n white vertices.

Original entry on oeis.org

1, 4, 38, 587, 12607, 347158, 11668113, 463118041, 21199488803, 1099465138203, 63715991036964, 4080500855334901, 286178278238641752, 21813909692571410084, 1795659553423061982001, 158754024731440581761116, 15002712207593790179795284, 1509215071938528737864389367, 161017605699030302902310357883

Offset: 1

Paul Laubie, Oct 21 2023

A Greg hypertree is a hypertree with black and white vertices, such that black vertices are unlabeled and have at least two incoming edges.

Cf. A005264, A052888.

Rest[CoefficientList[InverseSeries[Series[E^-x (1 + x + Log[1 + x]) - 1, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 24 2023 *)

my(t='t+O('t^25)); Vec(serlaplace(serreverse((log(1+t)-exp(t)+t+1)*exp(-t)))) \\ Michel Marcus, Oct 21 2023

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

a(n) ~ sqrt((1+s)*(2+s)/((1+r)*(3 + s*(3+s)))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where s = 0.3900539630495916058133890253422601894372373496844... is the root of the equation exp(-s + 1/(1+s)) = 1+s and r = exp(-s)*(1 + 1/(1+s)) - 1 = 0.1640664235584946357534702598223332293549130374395... - Vaclav Kotesovec, Oct 24 2023

A321231 Number of non-isomorphic connected weight-n multiset partitions with no singletons and multiset density -1.

Original entry on oeis.org

1, 0, 2, 3, 8, 15, 42, 94, 256, 656, 1807

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.

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) = 2 through a(5) = 15 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}
                      {{1,2},{2,2}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{3,3},{1,2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A317672, A318697, A321155, A321228, A321229.

A321255 Number of connected multiset partitions with multiset density -1, of strongly normal multisets of size n, with no singletons.

Original entry on oeis.org

0, 0, 2, 3, 8, 19, 60, 183, 643, 2355, 9393

Offset: 0

Gus Wiseman, Nov 01 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(2) = 2 through a(5) = 19 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,1,2}}  {{1,1,1,2}}    {{1,1,1,1,2}}
           {{1,2,3}}  {{1,1,2,2}}    {{1,1,1,2,2}}
                      {{1,1,2,3}}    {{1,1,1,2,3}}
                      {{1,2,3,4}}    {{1,1,2,2,3}}
                      {{1,1},{1,1}}  {{1,1,2,3,4}}
                      {{1,1},{1,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,1,2}}
                                     {{1,1},{1,2,2}}
                                     {{1,1},{1,2,3}}
                                     {{1,2},{1,1,1}}
                                     {{1,2},{1,1,3}}
                                     {{1,2},{1,3,4}}
                                     {{1,3},{1,1,2}}
                                     {{1,3},{1,2,2}}
                                     {{1,3},{1,2,4}}
                                     {{1,4},{1,2,3}}
                                     {{2,3},{1,1,2}}

Cf. A000272, A030019, A052888, A134954, A304867, A317672, A321228, A321229, A321253.

A357346 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(A(x)).

Original entry on oeis.org

0, 1, 5, 52, 849, 18996, 540986, 18726247, 763480675, 35837071558, 1903538106065, 112880374866172, 7392418912962210, 529898419942327801, 41266682731537698181, 3469461853041348996044, 313200848521114144611273, 30215925892728362737156556

Offset: 0

Seiichi Manyama, Sep 25 2022

nonn

Index entries for reversions of series

Cf. A052888, A357347, A357348, A357424.

Cf. A048802, A349557.

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

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

E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(-x)) ). - Seiichi Manyama, Sep 09 2024

A357348 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(3 * A(x)).

Original entry on oeis.org

0, 1, 9, 172, 5181, 214196, 11279542, 722242795, 54482959375, 4732518179422, 465226448603533, 51061919634063284, 6189640391474229790, 821277806639279795053, 118394082630978607655201, 18426248367244130561233924, 3079294928622816257125500821

Offset: 0

Seiichi Manyama, Sep 25 2022

nonn

Index entries for reversions of series

Cf. A052888, A357346, A357347, A357424.

Cf. A357345.

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

a(n) = Sum_{k=1..n} (n+3*k)^(k-1) * Stirling2(n,k).

E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(-3*x)) ). - Seiichi Manyama, Sep 09 2024

A357424 E.g.f. satisfies A(x) * exp(A(x)) = exp(x * exp(A(x))) - 1.

Original entry on oeis.org

0, 1, 1, 4, 21, 156, 1470, 16843, 227367, 3533974, 62163477, 1220852524, 26480355110, 628693388909, 16216901961481, 451609382251836, 13504072800481613, 431544662700594212, 14677503631085378170, 529370720888418692643, 20180856622352239827687

Offset: 0

Seiichi Manyama, Sep 27 2022

nonn

Index entries for reversions of series

Cf. A052888, A357346, A357347, A357348.

Cf. A349588.

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

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

a(n) = Sum_{k=1..n} (n-k)^(k-1) * Stirling2(n,k).
a(n) ~ n^(n-1) * (1 + exp(s)*s)^(n + 1/2) / (sqrt(exp(s)*(1 + s + s^2) - 1) * exp(n) * (1 + s)^(n - 1/2)), where s = 1.104072744884035178291292242554731... is the root of the equation 1 + s = (exp(-s) + s) * log(1 + exp(s)*s). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(x)) ). - Seiichi Manyama, Sep 09 2024

cp's OEIS Frontend

A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A322111 Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A357347 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(2 * A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A364709 Triangle read by rows: T(n,k) is the number of forests of labeled rooted hypertrees with n vertices and weight k, 0 <= k < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A364816 Number of labeled forests of rooted Greg hypertrees with n white vertices.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A321231 Number of non-isomorphic connected weight-n multiset partitions with no singletons and multiset density -1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A321255 Number of connected multiset partitions with multiset density -1, of strongly normal multisets of size n, with no singletons.

Views

Author

Keywords

Comments

Examples

Crossrefs

A357346 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A357348 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(3 * A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A357424 E.g.f. satisfies A(x) * exp(A(x)) = exp(x * exp(A(x))) - 1.

Views