A008827 -id:A008827 - cp's OEIS frontend

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

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

A002826 Number of precomplete Post functions of n variables.

Original entry on oeis.org

1, 5, 18, 82, 643, 15182, 7848984, 549761932909, 10626621620680478174719, 1701411834605079120446041612364090304458, 79607061350691085453966118726400345961810854094316840855510985236799831016092

Offset: 1

N. J. A. Sloane

nonn

S. V. Jablonskii, Some results in the theory of functional systems (Russian), in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 963-971, Acad. Sci. Fennica, Helsinki, 1980.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. Ju. Zaharova, V. B. Kudrjavcev, and S. V. Jablonskii, Precomplete classes in k-valued logics. (Russian) Dokl. Akad. Nauk SSSR 186 1969 509-512. English translation in Soviet Math. Doklady 10 (No. 3, 1969), 618-622.

Ivo Rosenberg, The number of maximal closed classes in the set of functions over a finite domain, J. Combinatorial Theory Ser. A 14 (1973), 1-7.
E. Ju. Zaharova, V. B. Kudrjavcev, and S. V. Jablonskii, Precomplete classes in k-valued logics. (Russian), Dokl. Akad. Nauk SSSR 186 (1969), 509-512. English translation in Soviet Math. Doklady 10 (No. 3, 1969), 618-622. [Annotated scanned copy]

Cf. A002824, A002825, A008827, A246069, A246137, A246417.

a(n) = A002824(n) + A246069(n) + A246137(n) + A008827(n) + A002825(n) + A246417(n). - Sean A. Irvine, Aug 25 2014

More terms from Sean A. Irvine, Aug 25 2014

A330667 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose atoms are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 1, 0, 1, 0, 1, 3, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 1, 5, 5, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 3, 0, 1, 1, 5, 9, 5, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 0, 1, 1, 0, 1, 0, 1, 5, 5, 0, 1, 1, 3

Offset: 1

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.

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.

			Triangle begins:
  {}
  1
  1
  1 0
  1
  1 0
  1
  1 1 0
  1 0
  1 0
  1
  1 2 0
  1
  1 0
  1 0
  1 3 2 0
  1
  1 2 0
  1
  1 2 0
Row n = 84 counts the following multisystems (commas elided):
  {1124}  {{1}{124}}    {{{1}}{{1}{24}}}
          {{11}{24}}    {{{11}}{{2}{4}}}
          {{12}{14}}    {{{1}}{{2}{14}}}
          {{2}{114}}    {{{12}}{{1}{4}}}
          {{4}{112}}    {{{1}}{{4}{12}}}
          {{1}{1}{24}}  {{{14}}{{1}{2}}}
          {{1}{2}{14}}  {{{2}}{{1}{14}}}
          {{1}{4}{12}}  {{{2}}{{4}{11}}}
          {{2}{4}{11}}  {{{24}}{{1}{1}}}
                        {{{4}}{{1}{12}}}
                        {{{4}}{{2}{11}}}

Row lengths are A001222.
Row sums are A318812.
The last nonzero term of row n is A330665(n).
Column k = 2 is 0 if n is prime; otherwise it is A001055(n) - 2.
Cf. A000311, A000669, A001678, A005121, A008827, A213427, A317145, A318846, A330474, A330475, A330655, A330666.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
totfac[n_,k_]:=If[k==1,1,Sum[totfac[Times@@Prime/@f,k-1],{f,Select[facs[n],1

A330936 Number of nontrivial factorizations of n into factors > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 1, 2, 0, 3, 0, 5, 0, 0, 0, 7, 0, 0, 0, 5, 0, 3, 0, 2, 2, 0, 0, 10, 0, 2, 0, 2, 0, 5, 0, 5, 0, 0, 0, 9, 0, 0, 2, 9, 0, 3, 0, 2, 0, 3, 0, 14, 0, 0, 2, 2, 0, 3, 0, 10, 3, 0, 0, 9, 0, 0

Offset: 1

Gus Wiseman, Jan 04 2020

nonn

The trivial factorizations of a number are (1) the case with only one factor, and (2) the factorization into prime numbers.

			The a(n) nontrivial factorizations of n = 8, 12, 16, 24, 36, 48, 60, 72:
  (2*4)  (2*6)  (2*8)    (3*8)    (4*9)    (6*8)      (2*30)    (8*9)
         (3*4)  (4*4)    (4*6)    (6*6)    (2*24)     (3*20)    (2*36)
                (2*2*4)  (2*12)   (2*18)   (3*16)     (4*15)    (3*24)
                         (2*2*6)  (3*12)   (4*12)     (5*12)    (4*18)
                         (2*3*4)  (2*2*9)  (2*3*8)    (6*10)    (6*12)
                                  (2*3*6)  (2*4*6)    (2*5*6)   (2*4*9)
                                  (3*3*4)  (3*4*4)    (3*4*5)   (2*6*6)
                                           (2*2*12)   (2*2*15)  (3*3*8)
                                           (2*2*2*6)  (2*3*10)  (3*4*6)
                                           (2*2*3*4)            (2*2*18)
                                                                (2*3*12)
                                                                (2*2*2*9)
                                                                (2*2*3*6)
                                                                (2*3*3*4)

Positions of nonzero terms are A033942.
Positions of 1's are A030078.
Positions of 2's are A054753.
Nontrivial integer partitions are A007042.
Nontrivial set partitions are A008827.
Nontrivial divisors are A070824.
Cf. A001055, A003238, A005121, A317145, A317176, A318812, A330665, A330935.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[DeleteCases[Rest[facs[n]],{_}]],{n,100}]

For prime n, a(n) = 0; for nonprime n, a(n) = A001055(n) - 2.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A002826 Number of precomplete Post functions of n variables.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A330667 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose atoms are the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A330936 Number of nontrivial factorizations of n into factors > 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula