cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-46 of 46 results.

A255959 Number of collections of nonempty multisets with a total of n+9 objects of exactly n colors.

Original entry on oeis.org

0, 42, 3836, 174652, 5579536, 144467430, 3265329848, 67312339598, 1301421849824, 24048952078889, 430422420403100, 7533527344234870, 129872860822059096, 2217253446393756056, 37645087319936852196, 637701933335948288525, 10805989785555925331616
Offset: 0

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

A diagonal of A255903.

Formula

a(n) = A255903(n+9,n).

A255960 Number of collections of nonempty multisets with a total of n+10 objects of exactly n colors.

Original entry on oeis.org

0, 56, 7045, 407367, 15939998, 492315490, 13017110235, 309183689612, 6804360339112, 141693764935536, 2833749745048664, 55021487525610716, 1045661876232247218, 19571912311804506186, 362535974017212548973, 6671022440871525234600, 122311996588305966744504
Offset: 0

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

A diagonal of A255903.

Formula

a(n) = A255903(n+10,n)

A258450 Number of collections of nonempty multisets of colored objects, where n is the number of objects plus the number of distinct colors.

Original entry on oeis.org

1, 0, 1, 2, 5, 13, 35, 100, 298, 926, 2995, 10045, 34871, 125040, 462283, 1759340, 6882479, 27639252, 113809750, 479993898, 2071411798, 9138568984, 41182104446, 189418562699, 888607018626, 4248949407337, 20695172225549, 102617378820155, 517728263280060
Offset: 0

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Examples

			a(4) = 5: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{2}}, {{1,2}}.

Links

Crossrefs

Antidiagonal sums of A255903.

Programs

  • Maple
    with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
      add(d*binomial(d+k-1, k-1), d=divisors(j)), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
a:= n-> add(T(n-i, i), i=0..n/2):
seq(a(n), n=0..30);
  • Mathematica
    A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k]*DivisorSum[j, #*Binomial[# +k-1, k-1]&], {j, 1, n}]/n];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := Sum[T[n-i, i], {i, 0, n/2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Formula

a(n) = Sum_{i=0..floor(n/2)} A255903(n-i,i).

A317178 Total number of colors summed over all collections of nonempty multisets with a total of n objects having color set {1,...,k} for some k <= n.

Original entry on oeis.org

0, 1, 6, 34, 210, 1378, 9726, 73314, 588846, 5018290, 45216798, 429290178, 4281260386, 44724792130, 488185773782, 5555082575762, 65760600166734, 808339570702354, 10299822335353654, 135828836289073570, 1851208197237464530, 26040184250745321058
Offset: 0

Views

Author

Alois P. Heinz, Jul 23 2018

Keywords

Links

Crossrefs

Cf. A255903.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*add(d*
      binomial(d+k-1, k-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k)*k, k=0..n):
seq(a(n), n=0..25);
  • Mathematica
    b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k]*Sum[d*Binomial[d + k - 1, k - 1], {d, Divisors[j]}], {j, 1, n}]/n];
a[n_] := Sum[Sum[b[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]*k, {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A255903(n,k).

A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 5, 8, 3, 1, 0, 1, 8, 18, 13, 3, 1, 0, 1, 9, 32, 37, 15, 3, 1, 0, 1, 13, 55, 96, 59, 16, 3, 1, 0, 1, 14, 91, 209, 196, 74, 16, 3, 1, 0, 1, 19, 138, 449, 573, 313, 82, 16, 3, 1, 0, 1, 20, 206, 863, 1529, 1147, 403, 84, 16, 3, 1
Offset: 0

Views

Author

Gus Wiseman, Dec 28 2023

Keywords

Comments

A set-system is a finite set of finite nonempty sets.
Conjecture: Column k = 2 is A101881.

Examples

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   1   4   3   1
   0   1   5   8   3   1
   0   1   8  18  13   3   1
   0   1   9  32  37  15   3   1
   0   1  13  55  96  59  16   3   1
   0   1  14  91 209 196  74  16   3   1
   0   1  19 138 449 573 313  82  16   3   1
   ...
Non-isomorphic representatives of the set-systems counted in row n = 5:
  .  {12345}  {1}{1234}  {1}{2}{123}  {1}{2}{3}{12}  {1}{2}{3}{4}{5}
              {1}{2345}  {1}{2}{134}  {1}{2}{3}{14}
              {12}{123}  {1}{2}{345}  {1}{2}{3}{45}
              {12}{134}  {1}{12}{13}
              {12}{345}  {1}{12}{23}
                         {1}{12}{34}
                         {1}{23}{24}
                         {1}{23}{45}

Links

Crossrefs

Row sums are A283877, connected case A300913.
For multiset partitions we have A317533.
Counting connected components instead of edges gives A321194.
For set multipartitions we have A334550.
For strict multiset partitions we have A368099.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A316980 counts non-isomorphic strict multiset partitions, connected A319557.
A319559 counts non-isomorphic T_0 set-systems, connected A319566.
Cf. A101881, A255903, A302545, A306005, A317532, A317794, A317795, A319560, A321405, A368094, A368095.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&And@@UnsameQ@@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]
  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

Extensions

Terms a(66) and beyond from Andrew Howroyd, Jan 11 2024

A368099 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 5, 1, 0, 7, 28, 22, 5, 1, 0, 11, 66, 83, 31, 5, 1, 0, 15, 134, 252, 147, 34, 5, 1, 0, 22, 280, 726, 620, 203, 35, 5, 1, 0, 30, 536, 1946, 2283, 1069, 235, 35, 5, 1, 0, 42, 1043, 4982, 7890, 5019, 1469, 248, 35, 5, 1
Offset: 0

Views

Author

Gus Wiseman, Dec 31 2023

Keywords

Examples

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    4    1
    0    5   12    5    1
    0    7   28   22    5    1
    0   11   66   83   31    5    1
    0   15  134  252  147   34    5    1
    0   22  280  726  620  203   35    5    1
    0   30  536 1946 2283 1069  235   35    5    1
    0   42 1043 4982 7890 5019 1469  248   35    5    1
    ...
Row n = 4 counts the following representatives:
  .  {{1,1,1,1}}  {{1},{1,1,1}}  {{1},{2},{1,1}}  {{1},{2},{3},{4}}
     {{1,1,1,2}}  {{1},{1,1,2}}  {{1},{2},{1,2}}
     {{1,1,2,2}}  {{1},{1,2,2}}  {{1},{2},{1,3}}
     {{1,1,2,3}}  {{1},{1,2,3}}  {{1},{2},{3,3}}
     {{1,2,3,4}}  {{1},{2,2,2}}  {{1},{2},{3,4}}
                  {{1},{2,2,3}}
                  {{1},{2,3,4}}
                  {{1,1},{1,2}}
                  {{1,1},{2,2}}
                  {{1,1},{2,3}}
                  {{1,2},{1,3}}
                  {{1,2},{3,4}}

Links

Crossrefs

Row sums are A316980, connected case A319557.
For multiset partitions we have A317533, connected A322133.
Counting connected components instead of edges gives A321194.
For normal multiset partitions we have A330787, row sums A317776.
For set multipartitions we have A334550.
For set-systems we have A368096, row-sums A283877 (connected A300913).
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
Cf. A255903, A296122, A302545, A306005, A317532, A317775, A317794, A317795, A319560, A368094, A368095.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024
Previous Showing 41-46 of 46 results.

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