A006472 -id:A006472 - cp's OEIS frontend

A331957 Number of rooted chains in set partitions of {1, 2, ..., n}.

1, 1, 2, 8, 64, 872, 18024, 525520, 20541392, 1036555120, 65591856032, 5085891210864, 474213645013904, 52346708185187392, 6751386193135966464, 1005991884967386086400, 171500271138273300946720, 33167303833191421470542496, 7222314392966179538774364128, 1759036134944451206655721276256

Offset: 0

S. R. Kannan and Rajesh Kumar Mohapatra, Feb 02 2020

nonn

Also the number of chains of Stirling numbers of the second kind such that the first term of the chains is either {{1}, {2}, ..., {n}} or {{1,2,...,n}}.

Number of rooted fuzzy equivalence matrices of order n.

			The a(3) = 8 in the lattice of set partitions of {1,2,3}:
  {{1},{2},{3}},
  {{1},{2},{3}} < {{1,2},{3}},
  {{1},{2},{3}} < {{1,3},{2}},
  {{1},{2},{3}} < {{1},{2,3}},
  {{1},{2},{3}} < {{1,2,3}},
  {{1},{2},{3}} < {{1,2},{3}} < {{1,2,3}},
  {{1},{2},{3}} < {{1,3},{2}} < {{1,2,3}},
  {{1},{2},{3}} < {{1},{2,3}} < {{1,2,3}}.
Or,
  {{1,2,3}},
  {{1,2,3}} > {{1,2},{3}},
  {{1,2,3}} > {{1,3},{2}},
  {{1,2,3}} > {{1},{2,3}},
  {{1,2,3}} > {{1},{2},{3}},
  {{1,2,3}} > {{1},{2,3}} > {{1},{2},{3}},
  {{1,2,3}} > {{2},{1,3}} > {{1},{2},{3}},
  {{1,2,3}} > {{3},{1,2}} > {{1},{2},{3}}.

Alois P. Heinz, Table of n, a(n) for n = 0..261
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
V. Murali, Equivalent finite fuzzy sets and Stirling numbers, Inf. Sci., 174 (2005), 251-263.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.

Row sums of A331956.
Cf. A000110, A058692, A006472.
Cf. A008277, A048993, A328044, A330301, A330302.
Cf. A086053.

b:= proc(n, k, t) option remember; `if`(k<0 or k>n, 0, `if`(k=1 or
      {n, k}={0}, 1, add(b(v, k-1, 1)*Stirling2(n, v), v=k..n-t)))
    end:
a:= n-> add(b(n, k, 0), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 09 2020

b[n_, k_, t_] := b[n, k, t] = If[k < 0 || k > n, 0, If[k == 1 || Union@{n, k} =={0}, 1, Sum[b[v, k - 1, 1]*StirlingS2[n, v], {v, k, n - t}]]];
a[n_] := Sum[b[n, k, 0], {k, 0, n}];
a /@ Range[0, 30]

b(n, k, t) = {if (k < 0, return(0)); if ((n==0) && (k==0), return (1)); if ((k==1) && (n>0), return(1)); sum(v = k, n - t, if (k==1, 1, b(v, k-1, 1))*stirling(n, v, 2));}
a(n) = sum(k=0, n, b(n, k, 0); ); \\ Michel Marcus, Feb 09 2020

from sympy.functions.combinatorial.numbers import stirling as s
from functools import cache
@cache
def a(n): return 1 + sum(s(n, k) * a(k) for k in range(1, n)) # David Radcliffe, Jul 01 2025

a(n) = Sum_{k=0..n} A331956(n,k).
Conjecture from Mikhail Kurkov, Jun 25 2025: (Start)
a(n) = R(n,0) where
  R(0,0) = 1,
  R(n,k) = (k+1) * Sum_{j=k..n-1} R(n-1,j) for 0 <= k < n,
  R(n,n) = Sum_{j=0..n-1} R(n,j). (End)
a(n) ~ A086053 * n!^2 / (2^(n-1) * log(2)^n * n^(1 + log(2)/3)). - Vaclav Kotesovec, Jul 01 2025
a(n) = 1 + Sum_{k=1..n-1} Stirling2(n,k)*a(k). - Rajesh Kumar Mohapatra, Jul 01 2025

More terms from Michel Marcus, Feb 08 2020

A330804 Number of chains in partitions of [n] ordered by refinement.

Original entry on oeis.org

1, 1, 3, 15, 127, 1743, 36047, 1051039, 41082783, 2073110239, 131183712063, 10171782421727, 948427290027807, 104693416370374783, 13502772386271932927, 2011983769934772172799, 343000542276546601893439, 66334607666382842941084991, 14444628785932359077548728255, 3518072269888902413311442552511

Offset: 0

S. R. Kannan, Rajesh Kumar Mohapatra, Jan 01 2020

nonn

Also the number of fuzzy equivalence matrices of order n.
Number of chains of equivalence relations on a set of n-elements.
Number of chains in Stirling numbers of the second kind.
Number of chains in the unordered partition of {1,...,n}.

			Consider the set S = {1, 2, 3}. The a(3) = 5+ 7+ 3 = 15 in the lattice of set partitions of {1,2,3}:
{{1},{2},{3}}  {{1},{2},{3}} < {{1,2},{3}}  {{1},{2},{3}} < {{1,2},{3}} < {{1,2,3}}
{{1,2},{3}}    {{1},{2},{3}} < {{1,3},{2}}  {{1},{2},{3}} < {{1,3},{2}} < {{1,2,3}}
{{1,3},{2}}    {{1},{2},{3}} < {{1},{2,3}}  {{1},{2},{3}} < {{1},{2,3}} < {{1,2,3}}
{{1},{2,3}}    {{1},{2},{3}} < {{1,2,3}}
{{1,2,3}}      {{1,2},{3}} < {{1,2,3}}
               {{1,3},{2}} < {{1,2,3}}
               {{1},{2,3}} < {{1,2,3}}

Alois P. Heinz, Table of n, a(n) for n = 0..262
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
V. Murali, Equivalent finite fuzzy sets and Stirling numbers, Inf. Sci., 174 (2005), 251-263.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.

Cf. A008277, A048993, A000110, A143494, A006472.
Cf. A007047, A328044, A330301, A330302, A331955.
Cf. A131407, A086053.

b:= proc(n, k, t) option remember; `if`(k<0, 0, `if`({n, k}={0}, 1,
      add(`if`(k=1, 1, b(v, k-1, 1))*Stirling2(n, v), v=k..n-t)))
    end:
a:= n-> add(b(n, k, 0), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 07 2020
# second Maple program:
a:= proc(n) option remember; uses combinat;
      bell(n) + add(stirling2(n, i)*a(i), i=1..n-1)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 03 2020

b[n_, k_, t_] := b[n, k, t] = If[k < 0, 0, If[Union@{n, k} == {0}, 1, Sum[If[k == 1, 1, b[v, k - 1, 1]]*StirlingS2[n, v], {v, k, n - t}]]];
a[n_] := Sum[b[n, k, 0], {k, 0, n}];
a /@ Range[0, 20] (* Jean-François Alcover, Feb 08 2020, after Alois P. Heinz *)

b(n, k, t) = {if (k < 0, return(0)); if ((n==0) && (k==0), return (1)); sum(v = k, n - t, if (k==1, 1, b(v, k-1, 1))*stirling(n, v, 2));}
a(n) = sum(k=0, n, b(n, k, 0);); \\ Michel Marcus, Feb 08 2020

a(n) = Sum_{k=0..n} A331955(n,k).
a(n) = Bell(n) + Sum_{i=1..n-1} Stirling2(n,i)*a(i). - Alois P. Heinz, Sep 03 2020
a(n) ~ A086053 * n!^2 / (2^(n-2) * log(2)^n * n^(1 + log(2)/3)). - Vaclav Kotesovec, Jul 01 2025
a(n) = 2 * A331957(n) - 1 = 4 * A005121(n) - 1 for n > 1. - Rajesh Kumar Mohapatra, Jul 01 2025

More terms from Michel Marcus, Feb 07 2020

A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.

Original entry on oeis.org

1, 1, 2, 4, 20, 140, 1411

Offset: 0

Gus Wiseman, Dec 27 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 20 multisystems:
  {1,1}  {{1},{1,1}}  {{{1}},{{1},{1,1}}}
  {1,2}  {{1},{1,2}}  {{{1,1}},{{1},{1}}}
         {{1},{2,3}}  {{{1}},{{1},{1,2}}}
         {{2},{1,1}}  {{{1,1}},{{1},{2}}}
                      {{{1}},{{1},{2,2}}}
                      {{{1,1}},{{2},{2}}}
                      {{{1}},{{1},{2,3}}}
                      {{{1,1}},{{2},{3}}}
                      {{{1}},{{2},{1,1}}}
                      {{{1,2}},{{1},{1}}}
                      {{{1}},{{2},{1,2}}}
                      {{{1,2}},{{1},{2}}}
                      {{{1}},{{2},{1,3}}}
                      {{{1,2}},{{1},{3}}}
                      {{{1}},{{2},{3,4}}}
                      {{{1,2}},{{3},{4}}}
                      {{{2}},{{1},{1,1}}}
                      {{{2}},{{1},{1,3}}}
                      {{{2}},{{3},{1,1}}}
                      {{{2,3}},{{1},{1}}}

The non-maximal version is A330474.
Labeled versions are A330675 (strongly normal) and A330676 (normal).
The case where the leaves are sets (as opposed to multisets) is A330677.
The case with all atoms distinct is A000111.
The case with all atoms equal is (also) A000111.
Cf. A000311, A004114, A005121, A006472, A007716, A048816, A141268, A306186, A330470, A330655, A330664.

A330675 Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 2, 6, 43, 440, 7158, 151414

Offset: 0

Gus Wiseman, Dec 30 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

			The a(2) = 2 and a(3) = 6 multisystems:
  {1,1}  {{1},{1,1}}
  {1,2}  {{1},{1,2}}
         {{1},{2,3}}
         {{2},{1,1}}
         {{2},{1,3}}
         {{3},{1,2}}
The a(4) = 43 multisystems (commas and outer brackets elided):
  {{1}}{{1}{11}} {{1}}{{1}{12}} {{1}}{{1}{22}} {{1}}{{1}{23}} {{1}}{{2}{34}}
  {{11}}{{1}{1}} {{11}}{{1}{2}} {{11}}{{2}{2}} {{11}}{{2}{3}} {{12}}{{3}{4}}
                 {{1}}{{2}{11}} {{1}}{{2}{12}} {{1}}{{2}{13}} {{1}}{{3}{24}}
                 {{12}}{{1}{1}} {{12}}{{1}{2}} {{12}}{{1}{3}} {{13}}{{2}{4}}
                 {{2}}{{1}{11}} {{2}}{{1}{12}} {{1}}{{3}{12}} {{1}}{{4}{23}}
                                {{2}}{{2}{11}} {{13}}{{1}{2}} {{14}}{{2}{3}}
                                {{22}}{{1}{1}} {{2}}{{1}{13}} {{2}}{{1}{34}}
                                               {{2}}{{3}{11}} {{2}}{{3}{14}}
                                               {{23}}{{1}{1}} {{23}}{{1}{4}}
                                               {{3}}{{1}{12}} {{2}}{{4}{13}}
                                               {{3}}{{2}{11}} {{24}}{{1}{3}}
                                                              {{3}}{{1}{24}}
                                                              {{3}}{{2}{14}}
                                                              {{3}}{{4}{12}}
                                                              {{34}}{{1}{2}}
                                                              {{4}}{{1}{23}}
                                                              {{4}}{{2}{13}}
                                                              {{4}}{{3}{12}}

The case with all atoms equal is A000111.
The case with all atoms different is A006472.
The version allowing all depths is A330475.
The unlabeled version is A330663.
The version where the atoms are the prime indices of n is A330665.
The (weakly) normal version is A330676.
The version where the degrees are the prime indices of n is A330728.
Multiset partitions of strongly normal multisets are A035310.
Series-reduced rooted trees with strongly normal leaves are A316652.
Cf. A000311, A000669, A001055, A001678, A005121, A005804, A316651, A318812, A330467, A330474, A330625, A330628, A330664, A330677, A330679.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 16, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 16, 1, 3, 1, 2, 1, 3, 1, 27, 1, 1, 2, 2, 1, 3, 1, 16, 2, 1, 1, 11, 1

Offset: 1

Gus Wiseman, Dec 27 2019

nonn

First differs from A317145 at a(32) = 5, A317145(32) = 4.
A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also series/singleton-reduced factorizations of n with Omega(n) levels of parentheses. See A001055, A050336, A050338, A050340, etc.

			The a(n) multisystems for n = 2, 6, 12, 24, 48:
  {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}  {{{{1}}},{{{1}},{{1},{1,2}}}}
              {{2},{1,1}}  {{{1,1}},{{1},{2}}}  {{{{1}}},{{{1,1}},{{1},{2}}}}
                           {{{1}},{{2},{1,1}}}  {{{{1},{1}}},{{{1}},{{1,2}}}}
                           {{{1,2}},{{1},{1}}}  {{{{1},{1,1}}},{{{1}},{{2}}}}
                           {{{2}},{{1},{1,1}}}  {{{{1,1}}},{{{1}},{{1},{2}}}}
                                                {{{{1}}},{{{1}},{{2},{1,1}}}}
                                                {{{{1}}},{{{1,2}},{{1},{1}}}}
                                                {{{{1},{1}}},{{{2}},{{1,1}}}}
                                                {{{{1},{1,2}}},{{{1}},{{1}}}}
                                                {{{{1,1}}},{{{2}},{{1},{1}}}}
                                                {{{{1}}},{{{2}},{{1},{1,1}}}}
                                                {{{{1},{2}}},{{{1}},{{1,1}}}}
                                                {{{{1,2}}},{{{1}},{{1},{1}}}}
                                                {{{{2}}},{{{1}},{{1},{1,1}}}}
                                                {{{{2}}},{{{1,1}},{{1},{1}}}}
                                                {{{{2},{1,1}}},{{{1}},{{1}}}}

The last nonzero term in row n of A330667 is a(n).
The chain version is A317145.
The non-maximal version is A318812.
Unlabeled versions are A330664 and A330663.
Other labeled versions are A330675 (strongly normal) and A330676 (normal).
Cf. A001055, A005121, A005804, A050336, A213427, A292505, A317144, A318849, A320160, A330474, A330475, A330679.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

a(2^n) = A000111(n - 1).

a(product of n distinct primes) = A006472(n).

A330676 Number of balanced reduced multisystems of weight n and maximum depth whose atoms cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 8, 70, 1012, 21944, 665708, 26917492, 1399033348, 90878863352, 7214384973908, 687197223963640, 77354805301801012, 10158257981179981304, 1539156284259756811748, 266517060496258245459352, 52301515332984084095078308, 11546416513975694879642736152

Offset: 0

Gus Wiseman, Dec 30 2019

nonn

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

A finite multiset is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 8 multisystems:
  {}  {1}  {1,1}  {{1},{1,1}}
           {1,2}  {{1},{1,2}}
                  {{1},{2,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,2}}
                  {{2},{1,3}}
                  {{3},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A330778.
The case with all atoms equal is A000111.
The case with all atoms different is A006472.
The version allowing all depths is A330655.
The unlabeled version is A330663.
The version where the atoms are the prime indices of n is A330665.
The strongly normal version is A330675.
The version where the degrees are the prime indices of n is A330728.
Multiset partitions of normal multisets are A255906.
Series-reduced rooted trees with normal leaves are A316651.
Cf. A000669, A001055, A005121, A005804, A318812, A330469, A330474, A330654, A330664, A330677, A330679.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, for(i=n, #v, u[i] += v[i]*(-1)^(i-n)*binomial(i-1, n-1)); v=EulerT(v)); u}
seq(n)={concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2019

A010796 a(n) = n!*(n+1)!/2.

Original entry on oeis.org

1, 6, 72, 1440, 43200, 1814400, 101606400, 7315660800, 658409472000, 72425041920000, 9560105533440000, 1491376463216640000, 271430516305428480000, 57000408424139980800000, 13680098021793595392000000, 3720986661927857946624000000

Offset: 1

N. J. A. Sloane

Column 2 in triangle A009963.
a(n) = A078740(n, 2), first column of (3, 2)-Stirling2 array.
Also the number of undirected Hamiltonian paths in the complete bipartite graph K_{n,n+1}. - Eric W. Weisstein, Sep 03 2017
Also, the number of undirected Hamiltonian cycles in the complete bipartite graph K_{n+1,n+1}. - Pontus von Brömssen, Sep 06 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Index entries for sequences related to factorial numbers.

Cf. A006472, A009963, A078740, A010790.

Main diagonal of A291909.

[Factorial(n)* Factorial(n+1) / 2: n in [1..20]]; // Vincenzo Librandi, Jun 11 2013

Table[n! (n + 1)! / 2, {n, 1, 20}] (* Vincenzo Librandi, Jun 11 2013 *)
Times@@@Partition[Range[20]!,2,1]/2 (* Harvey P. Dale, Jul 04 2017 *)

for(n=1,30, print1(n!*(n+1)!/2, ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = 2^(n-1) * A006472(n+1).
a(n) = A010790(n)/2.
E.g.f.: (hypergeom([1, 2], [], x)-1)/2.
a(n) = Product_{k=1..n-1} (k^2+3*k+2). - Gerry Martens, May 09 2016
E.g.f.: x*hypergeom([1, 3], [], x). - Robert Israel, May 09 2016
From Amiram Eldar, Jun 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(BesselI(1, 2) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - BesselJ(1, 2)). (End)

A008826 Triangle of coefficients from fractional iteration of e^x - 1.

Original entry on oeis.org

1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, 875, 16674, 74165, 114345, 56700, 1, 4138, 155477, 1208830, 3394790, 3919860, 1587600, 1, 21145, 1542699, 19800165, 90265560, 182184030, 167310360, 57153600, 1, 115973, 16385857, 335976195, 2338275240, 7342024200, 11471572350, 8719666200, 2571912000

Offset: 2

N. J. A. Sloane, Mar 15 1996

The triangle reflects the Jordan-decomposition of the matrix of Stirling numbers of the second kind. A display of the matrix formula can be found at the Helms link which also explains the generation rule for the A()-numbers in a different way. - Gottfried Helms Apr 19 2014
From Gus Wiseman, Jan 02 2020: (Start)
Also the number of balanced reduced multisystems with atoms {1..n} and depth k. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. For example, row n = 4 counts the following multisystems:
  {1,2,3,4}  {{1},{2,3,4}}    {{{1}},{{2},{3,4}}}
             {{1,2},{3,4}}    {{{1},{2}},{{3,4}}}
             {{1,2,3},{4}}    {{{1},{2,3}},{{4}}}
             {{1,2,4},{3}}    {{{1,2}},{{3},{4}}}
             {{1,3},{2,4}}    {{{1,2},{3}},{{4}}}
             {{1,3,4},{2}}    {{{1},{2,4}},{{3}}}
             {{1,4},{2,3}}    {{{1,2},{4}},{{3}}}
             {{1},{2},{3,4}}  {{{1}},{{3},{2,4}}}
             {{1},{2,3},{4}}  {{{1},{3}},{{2,4}}}
             {{1,2},{3},{4}}  {{{1,3}},{{2},{4}}}
             {{1},{2,4},{3}}  {{{1,3},{2}},{{4}}}
             {{1,3},{2},{4}}  {{{1},{3,4}},{{2}}}
             {{1,4},{2},{3}}  {{{1,3},{4}},{{2}}}
                              {{{1}},{{4},{2,3}}}
                              {{{1},{4}},{{2,3}}}
                              {{{1,4}},{{2},{3}}}
                              {{{1,4},{2}},{{3}}}
                              {{{1,4},{3}},{{2}}}
(End)
From Harry Richman, Mar 30 2023: (Start)
Equivalently, T(n,k) is the number of length-k chains from minimum to maximum in the lattice of set partitions of {1..n} ordered by refinement. For example, row n = 4 counts the following chains, leaving out the minimum {1|2|3|4} and maximum {1234}:
  (empty)  {12|3|4}  {12|3|4} < {123|4}
           {13|2|4}  {12|3|4} < {124|3}
           {14|2|3}  {12|3|4} < {12|34}
           {1|23|4}  {13|2|4} < {123|4}
           {1|24|3}  {13|2|4} < {134|2}
           {1|2|34}  {13|2|4} < {13|24}
           {123|4}   {14|2|3} < {124|3}
           {124|3}   {14|2|3} < {134|2}
           {134|2}   {14|2|3} < {14|23}
           {1|234}   {1|23|4} < {123|4}
           {12|34}   {1|23|4} < {1|234}
           {13|24}   {1|23|4} < {14|23}
           {14|23}   {1|24|3} < {124|3}
                     {1|24|3} < {1|234}
                     {1|24|3} < {13|24}
                     {1|2|34} < {134|2}
                     {1|2|34} < {1|234}
                     {1|2|34} < {12|34}
(End)
Also the number of cells of dimension k in the fine subdivision of the Bergman complex of the complete graph on n vertices. - Harry Richman, Mar 30 2023

			Triangle starts:
  1;
  1,    3;
  1,   13,     18;
  1,   50,    205,     180;
  1,  201,   1865,    4245,    2700;
  1,  875,  16674,   74165,  114345,   56700;
  1, 4138, 155477, 1208830, 3394790, 3919860, 1587600;
  ...
The f-vector of (the fine subdivision of) the Bergman complex of the complete graph K_3 is (1, 3). The f-vector of the Bergman complex of K_4 is (1, 13, 18). - _Harry Richman_, Mar 30 2023

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.

Alois P. Heinz, Rows n = 2..150, flattened (first 19 rows from Vincenzo Librandi)
Gottfried Helms, How this expression leads to the given sequence, MathOverflow.
Federico Ardila and Caroline J. Klivans, The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B, 96 (2006), 38-49.

Row sums are A005121.
Alternating row sums are signed factorials A133942(n-1).
Column k = 2 is A008827.
Diagonal k = n - 1 is A006472.
Diagonal k = n - 2 is A059355.
Row n equals row 2^n of A330727.
Cf. A000110, A000111, A000258, A002846, A005121, A008277, A306186, A317176, A318813, A320154, A330667, A330679, A330784.

b:= proc(n) option remember; expand(`if`(n=1, 1,
      add(Stirling2(n, j)*b(j)*x, j=0..n-1)))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=1..n-1), n=2..10);  # Alois P. Heinz, Mar 31 2023

a[n_, x_] := Sum[ StirlingS2[n, k]*a[k, x]*x, {k, 0, n-1}]; a[1, ] = 1; Table[ CoefficientList[ a[n, x], x] // Rest, {n, 2, 10}] // Flatten (* _Jean-François Alcover, Dec 11 2012, after Vladeta Jovovic *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
tots[m_]:=Prepend[Join@@Table[tots[p],{p,Select[sps[m],1Gus Wiseman, Jan 02 2020 *)

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} Stirling2(n, k)*A(k;x)*x, A(1;x) = 1. - Vladeta Jovovic, Jan 02 2004

Sum_{k=1..n-1} (-1)^k*T(n,k) = (-1)^(n-1)*(n-1)! = A133942(n-1). - Geoffrey Critzer, Sep 06 2020

More terms from Vladeta Jovovic, Jan 02 2004

A087047 a(n) = n(n+1)(n+2)*a(n-1)/6 for n >= 1; a(0) = 1.

Original entry on oeis.org

1, 1, 4, 40, 800, 28000, 1568000, 131712000, 15805440000, 2607897600000, 573737472000000, 164088916992000000, 59728365785088000000, 27176406432215040000000, 15218787602040422400000000, 10348775569387487232000000000, 8444600864620189581312000000000, 8182818237816963704291328000000000

Offset: 0

Enrico T. Federighi (rico125162(AT)aol.com), Aug 08 2003

Product of the first n tetrahedral (or pyramidal) numbers. See 2nd formula. - Alexander Adamchuk, May 19 2006
From Peter Bala, Nov 28 2024: (Start)
For n >= 5, a(n-3) == 9 (mod n) if and only if n is a prime (adapt the proof of the Main Theorem in Himane).
The list of primes p such that a(p-3) == 9 (mod p^2) (analog of A007540 - Wilson primes) begins [11, 31, 47, ...]. (End)

			a(4) = (1/32)*(1/81)*24*120*720 = 800.

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
Djamel Himane, A simple proof of Werner Schulte's conjecture, arXiv:2404.08646 [math.GM], 2024
Ana Luzón, Manuel A. Morón, and José L. Ramírez, Differential Equations in Ward's Calculus, ResearchGate, September 2023.
Eric Weisstein's World of Mathematics, Tetrahedral Number.

Cf. A000292, A006472.

a[0]:=1: for n from 1 to 20 do a[n]:=n*(n+1)*(n+2)*a[n-1]/6 od: seq(a[n],n=0..17); # Emeric Deutsch, Mar 06 2005
seq(mul(binomial(k+2, 3), k=1..n), n=0..16); # Zerinvary Lajos, Aug 07 2007

Table[Product[k*(k+1)*(k+2)/6,{k,1,n}],{n,0,16}] (* Alexander Adamchuk, May 19 2006 *)
a[n_]:=Denominator[SeriesCoefficient[HypergeometricPFQ[{1},{1,2,3},6x],{x,0,n}]]; Array[a,18,0] (* Stefano Spezia, Oct 13 2023 *)

q=50 # change q for more terms
[2^(-n-1)*3^(-n)*factorial(n)*factorial(n+1)*factorial(n+2) for n in [0..q]] # Tom Edgar, Mar 15 2014

a(n) = 2^(-n-1)*3^(-n)*n!*(n+1)!*(n+2)!.
From Alexander Adamchuk, May 19 2006: (Start)
a(n) = Product_{k=1..n} k*(k+1)*(k+2)/6.
a(n) = Product_{k=1..n} A000292(k). (End)
a(n) = denominator( [x^n] 1F3([1], [1, 2, 3], 6*x) ), where 1F3 is the hypergeometric function (see Luzón et al. at page 19). - Stefano Spezia, Oct 13 2023

More terms from Emeric Deutsch, Mar 06 2005
Example and formula corrected by Tom Edgar, Mar 15 2014
a(0)=1 prepended by and a(15)-a(17) from Stefano Spezia, Oct 13 2023

A331955 Triangle T(n,k) of number of chains of length k in partitions of an n-set ordered by refinement.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 7, 3, 0, 15, 45, 49, 18, 0, 52, 306, 640, 565, 180, 0, 203, 2268, 8176, 13055, 9645, 2700, 0, 877, 18425, 108388, 279349, 359555, 227745, 56700, 0, 4140, 163754, 1523922, 5967927, 11918270, 12822110, 7095060, 1587600

Offset: 0

S. R. Kannan, Rajesh Kumar Mohapatra, Feb 02 2020

Also the number of chains of equivalence relations of length k on a set of n-points.
Number of chains of length k in Stirling numbers of the second kind.
Number of chains of length k in the unordered partition of {1,2,...,n}.
Number of k-level fuzzy equivalence matrices of order n.

			The triangle T(n,k) begins:
  n\k 0   1     2      3      4       5     6     7...
  0   1
  1   0   1
  2   0   2     1
  3   0   5     7      3
  4   0  15    45     49     18
  5   0  52   306    640    565    180
  6   0 203  2268   8176  13055   9645   2700
  7   0 877 18425 108388 279349 359555 227745 56700
  ...
The T(3,2) = 7 in the lattice of set partitions of {1,2,3}:
  {{1},{2},{3}} < {{1,2},{3}},
  {{1},{2},{3}} < {{1,3},{2}},
  {{1},{2},{3}} < {{1},{2,3}},
  {{1},{2},{3}} < {{1,2,3}},
  {{1,2},{3}} < {{1,2,3}},
  {{1,3},{2}} < {{1,2,3}},
  {{1},{2,3}} < {{1,2,3}}.

Alois P. Heinz, Rows n = 0..140, flattened
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
V. Murali, Equivalent finite fuzzy sets and Stirling numbers, Inf. Sci., 174 (2005), 251-263.
V. Murali, Combinatorics of counting finite fuzzy subsets, Fuzzy Sets Syst., 157(17)(2006), 2403-2411.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.

Cf. A000007 (column k=0), A000110 (column k=1), A006472 (diagonal), A330804 (row sums).
T(2n,n) gives A332244.
Cf. A008277, A048993, A328044, A330301, A330302.

b:= proc(n, k, t) option remember; `if`(k<0, 0, `if`({n, k}={0}, 1,
      add(`if`(k=1, 1, b(v, k-1, 1))*Stirling2(n, v), v=k..n-t)))
    end:
T:= (n, k)-> b(n, k, 0):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 07 2020

b[n_, k_, t_] := b[n, k, t] = If[k < 0, 0, If[Union@{n, k} == {0}, 1, Sum[If[k == 1, 1, b[v, k - 1, 1]]*StirlingS2[n, v], {v, k, n - t}]]];
T[n_, k_] := b[n, k, 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 08 2020, after Alois P. Heinz *)
T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[(n == 0 && k == 0), 1, If[k == 1, BellB[n], Sum[If[r >= 0, StirlingS2[n, r]*T[r, k - 1], 0], {r, k - 1, n - 1}]]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Rajesh Kumar Mohapatra, Jul 02 2025 *)

b(n, k, t) = {if (k < 0, return(0)); if ((n==0) && (k==0), return (1)); sum(v = k, n - t, if (k==1, 1, b(v, k-1, 1))*stirling(n, v, 2));}
T(n, k) = b(n, k, 0);
matrix(8, 8, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Feb 08 2020

T(0, 0) = 1, T(0, k) = 0 for k > 0.
T(n, k) = Sum_{i_k=k..n} (Sum_{i_(k-1)=k-1..i_k - 1} (... (Sum_{i_2=2..i_3 - 1} (Sum_{i_1=1..i_2 - 1} Stirling2(n,i_k) * Stirling2(i_k,i_(k-1)) * ... * Stirling2(i_3,i_2) * Stirling2(i_2,i_1)))...)), where 1 <= k <= n.
T(n,k) = Sum_{j=k-1..n-1} Stirling2(n,j)*T(j,k-1), 2 <= k <= n; T(n,1) = Bell(n), n >= 1; T(n,0) = A000007(n). - Rajesh Kumar Mohapatra, Jul 01 2025

cp's OEIS Frontend

A331957 Number of rooted chains in set partitions of {1, 2, ..., n}.

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A330804 Number of chains in partitions of [n] ordered by refinement.

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A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.

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A330675 Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n.

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A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

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A330676 Number of balanced reduced multisystems of weight n and maximum depth whose atoms cover an initial interval of positive integers.

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A010796 a(n) = n!*(n+1)!/2.

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A087047 a(n) = n(n+1)(n+2)*a(n-1)/6 for n >= 1; a(0) = 1.