A005121 -id:A005121 - cp's OEIS frontend

A330776 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with atoms colored using exactly k colors.

1, 1, 1, 2, 6, 4, 6, 37, 63, 32, 20, 262, 870, 1064, 436, 90, 2217, 12633, 27824, 26330, 9012, 468, 21882, 201654, 710712, 1163320, 895608, 262760, 2910, 249852, 3578610, 18924846, 47608000, 61786254, 40042128, 10270696, 20644, 3245520, 70539124, 538018360, 1950556400, 3792461176, 4070160416, 2275829088, 518277560

Offset: 1

Andrew Howroyd, Dec 30 2019

See A330655 for the definition of a balanced reduced multisystem.

A balanced reduced multisystem of weight n with atoms of k colors corresponds with a rooted tree with n leaves of k colors with all leaves at the same depth and at least one node at each level of the tree having more than one child. The final condition is needed to ensure that the number of such trees is finite.

			Triangle begins:
    1;
    1,     1;
    2,     6,      4;
    6,    37,     63,     32;
   20,   262,    870,   1064,     436;
   90,  2217,  12633,  27824,   26330,   9012;
  468, 21882, 201654, 710712, 1163320, 895608, 262760;
  ...
The T(3,2) = 6 balanced reduced multisystems are: {1,1,2}, {1,2,2}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}.

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

Column 1 is A318813.
Main diagonal is A005121.
Row sums are A330655.

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, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-1,n-1)); v=EulerT(v)); u}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))}

A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 9, 28, 36, 16, 1, 13, 69, 160, 164, 61, 1, 20, 160, 580, 1022, 855, 272, 1, 28, 337, 1837, 4996, 7072, 4988, 1385

Offset: 2

Gus Wiseman, Jan 03 2020

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.

			Triangle begins:
    1
    1    1
    1    3    2
    1    5    9    5
    1    9   28   36   16
    1   13   69  160  164   61
    1   20  160  580 1022  855  272
    1   28  337 1837 4996 7072 4988 1385
Row n = 5 counts the following multisystems (strings of 1's are replaced by their lengths):
  5  {1,4}      {{1},{1,3}}      {{{1}},{{1},{1,2}}}
     {2,3}      {{1},{2,2}}      {{{1,1}},{{1},{2}}}
     {1,1,3}    {{2},{1,2}}      {{{1}},{{2},{1,1}}}
     {1,2,2}    {{3},{1,1}}      {{{1,2}},{{1},{1}}}
     {1,1,1,2}  {{1},{1,1,2}}    {{{2}},{{1},{1,1}}}
                {{1,1},{1,2}}
                {{2},{1,1,1}}
                {{1},{1},{1,2}}
                {{1},{2},{1,1}}

Row sums are A318813.
Column k = 3 is A007042.
Column k = 4 is A001970(n) - 3*A000041(n) + 3.
Column k = n is A000111.
Row n is row prime(n) of A330727.
Cf. A000669, A001055, A002846, A005121, A196545, A213427, A318812, A320160, A330474, A330475, A330655, A330667, A330679.

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

T(n,3) = A000041(n) - 2.

T(n,4) = A001970(n) - 3 * A000041(n) + 3.

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.

A385521 Decimal expansion of a constant related to A375838.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jul 01 2025

Variant of Lengyel's constant A086053.

			1.59585433050036621247006569740016516964502505848324064247941890934119103861277...

Cf. A086053, A260932.
Cf. A086555, A246040, A375838.
Cf. A005121, A331957, A131407, A330804.

Equals lim_{n->oo} A375838(n) * 2^n * log(2)^n * n^(1-log(2)/3) / n!^2.

A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

Original entry on oeis.org

1, 1, 2, 3, 7, 17, 54, 199, 869, 4341, 24514, 154187

Offset: 0

Gus Wiseman, Jan 03 2020

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 a(1) = 1 through a(5) = 17 multisystems (commas elided):
  {1}  {2}   {3}        {4}               {5}
       {11}  {12}       {22}              {23}
             {{1}{11}}  {13}              {14}
                        {{1}{12}}         {{1}{13}}
                        {{2}{11}}         {{1}{22}}
                        {{{1}}{{1}{11}}}  {{2}{12}}
                        {{{11}}{{1}{1}}}  {{3}{11}}
                                          {{{1}}{{1}{12}}}
                                          {{{11}}{{1}{2}}}
                                          {{{1}}{{2}{11}}}
                                          {{{12}}{{1}{1}}}
                                          {{{2}}{{1}{11}}}
                                          {{{{1}}}{{{1}}{{1}{11}}}}
                                          {{{{1}}}{{{11}}{{1}{1}}}}
                                          {{{{1}{1}}}{{{1}}{{11}}}}
                                          {{{{1}{11}}}{{{1}}{{1}}}}
                                          {{{{11}}}{{{1}}{{1}{1}}}}

The case with all atoms equal to 1 is A000111.
The non-maximal version is A330679.
A tree version is A320160.
Cf. A000669, A002846, A005121, A141268, A196545, A213427, A317145, A318813, A330663, A330665, A330675, A330676, A330728.

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

A308476 a(1) = 1; a(n) = Sum_{k=1..n-1, gcd(n,k) = 1} Stirling2(n,k)*a(k).

Original entry on oeis.org

1, 1, 4, 25, 366, 5491, 176569, 5332097, 276268942, 13470365431, 1135683784753, 75066413338423, 9256260956838520, 918768523598548169, 140268128758724744770, 18398287904991375995745, 3879391299475140314514162, 594721341754741064012714341

Offset: 1

Ilya Gutkovskiy, May 29 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A005121, A045545, A308463.

a := proc(n) local j; option remember;
if n = 1 then 1;
else add(`if`(gcd(n, j) = 1, Stirling2(n, j)*a(j), 0), j = 1 .. n - 1);
end if; end proc;
seq(a(n), n = 1 .. 30); # G. C. Greubel, Mar 08 2021

a[n_] := Sum[If[GCD[n, k] == 1, StirlingS2[n, k] a[k], 0], {k, 1, n - 1}]; a[1] = 1; Table[a[n], {n, 1, 18}]

@CachedFunction
def a(n):
    if n==1: return 1
    else: return sum( stirling_number2(n,j)*a(j) if gcd(n,j)==1 else 0 for j in (1..n-1) )
[a(n) for n in (1..30)] # G. C. Greubel, Mar 08 2021

A330785 Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 8, 4, 0, 1, 9, 25, 28, 11, 0, 1, 13, 57, 111, 99, 33, 0, 1, 20, 129, 379, 561, 408, 116, 0, 1, 28, 253, 1057, 2332, 2805, 1739, 435, 0, 1, 40, 496, 2833, 8695, 15271, 15373, 8253, 1832, 0, 1, 54, 898, 6824, 28071, 67790, 98946, 85870, 40789, 8167

Offset: 1

Gus Wiseman, Jan 03 2020

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   3   2
   0   1   5   8   4
   0   1   9  25  28  11
   0   1  13  57 111  99  33
   0   1  20 129 379 561 408 116
Row n = 5 counts the following chains (minimum and maximum not shown):
  ()  (14)    (113)->(14)    (1112)->(113)->(14)
      (23)    (113)->(23)    (1112)->(113)->(23)
      (113)   (122)->(14)    (1112)->(122)->(14)
      (122)   (122)->(23)    (1112)->(122)->(23)
      (1112)  (1112)->(14)
              (1112)->(23)
              (1112)->(113)
              (1112)->(122)

Row sums are A213427.
Main diagonal is A002846.
Column k=3 is A007042.
Dominated by A330784.
The version for set partitions is A008826.
The version for factorizations is A330935.
Cf. A000111, A000258, A000311, A005121, A141268, A196545, A265947, A300383, A306186, A317141, A317176, A318813, A320160, A330679.

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]]]];
upr[q_]:=Union[Sort/@Apply[Plus,mps[q],{2}]];
paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y,#}&/@DeleteCases[upr[y],y],{y,Sort/@IntegerPartitions[n]}],ConstantArray[1,n],{n}],Length[#]==k-1&]],{n,8},{k,n}]

T(n,k) = A330935(2^n,k).

A246307 Numerator of Z^(2)(n), where Z^(2)(n) = n for n=0,1; thereafter Z^(2)(n) = (1/3)Sum_{k=1..n-1} Stirling_2(n,k)Z^(2)(k).

Original entry on oeis.org

0, 1, 1, 5, 6, 399, 10137, 364737, 2206077, 276269667, 21732613641, 2097942773859, 60958311638283, 16792338947372883, 2704712327221326273, 503673752669173980741, 6711263837756846638875, 3248087145389524173611367, 885435154962504420364992693, 270090359296255369532260168299

Offset: 0

N. J. A. Sloane, Aug 22 2014

nonn

The denominators are various powers of 2.

			The sequence Z^(2)(n) begins 0, 1, 1/2, 5/4, 6, 399/8, 10137/16, 364737/32, 2206077/8, 276269667/32,  21732613641/64, 2097942773859/128, 60958311638283/64, 16792338947372883/256,  2704712327221326273/512,...

D. Barsky, J.-P. Bézivin, p-adic Properties of Lengyel's Numbers, Journal of Integer Sequences, 17 (2014), #14.7.3.

Cf. A005121.

Z:=proc(n,p) option remember; local k; if n <= 1 then n else add(stirling2(n,k)*Z(k,p)/(p-1),k=1..n-1); fi; end;
t1:=[seq(Z(n,2),n=0..35)];

cp's OEIS Frontend

A330776 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with atoms colored using exactly k colors.

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PARI

A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

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

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A385521 Decimal expansion of a constant related to A375838.

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A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

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A308476 a(1) = 1; a(n) = Sum_{k=1..n-1, gcd(n,k) = 1} Stirling2(n,k)*a(k).

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Sage

A330785 Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.

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A246307 Numerator of Z^(2)(n), where Z^(2)(n) = n for n=0,1; thereafter Z^(2)(n) = (1/3)*Sum_{k=1..n-1} Stirling_2(n,k)*Z^(2)(k).

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Maple

A246307 Numerator of Z^(2)(n), where Z^(2)(n) = n for n=0,1; thereafter Z^(2)(n) = (1/3)Sum_{k=1..n-1} Stirling_2(n,k)Z^(2)(k).