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.

Showing 1-10 of 46 results. Next

A255906 Number of collections of nonempty multisets with a total of n objects having color set {1,...,k} for some k<=n.

Original entry on oeis.org

1, 1, 4, 16, 76, 400, 2356, 15200, 106644, 806320, 6526580, 56231024, 513207740, 4941362512, 50013751812, 530481210672, 5880285873060, 67954587978448, 816935340368068, 10196643652651664, 131904973822724540, 1765645473517011568, 24420203895517396180
Offset: 0

Views

Author

Alois P. Heinz, Mar 10 2015

Keywords

Comments

Number of multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Jul 30 2018

Examples

			a(0) = 1: {}.
a(1) = 1: {{1}}.
a(2) = 4: {{1},{1}}, {{1,1}}, {{1},{2}}, {{1,2}}.
a(3) = 16: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{1},{2}}, {{1},{2},{2}}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}, {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}.

Links

Crossrefs

Row sums of A255903. Also row sums of A317532.
Cf. A007716, A034691, A116539, A317073.

Programs

  • Mathematica
    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]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Join@@mps/@allnorm[n]],{n,6}] (* Gus Wiseman, Jul 30 2018 *)
  • PARI
    R(n, k)={Vec(-1 + 1/prod(j=1, n, (1 - x^j + O(x*x^n))^binomial(k+j-1, j) ))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

Formula

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

A317533 Regular triangle read rows: T(n,k) = number of non-isomorphic multiset partitions of size n and length k.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 14, 9, 5, 7, 28, 33, 16, 7, 11, 69, 104, 74, 29, 11, 15, 134, 294, 263, 142, 47, 15, 22, 285, 801, 948, 599, 263, 77, 22, 30, 536, 2081, 3058, 2425, 1214, 453, 118, 30, 42, 1050, 5212, 9769, 9276, 5552, 2322, 761, 181, 42, 56, 1918, 12645, 29538, 34172, 23770, 11545, 4179, 1223, 267, 56
Offset: 1

Views

Author

Gus Wiseman, Jul 30 2018

Keywords

Examples

			Non-isomorphic representatives of the T(3,2) = 4 multiset partitions:
  {{1},{1,1}}
  {{1},{1,2}}
  {{1},{2,2}}
  {{1},{2,3}}
Triangle begins:
    1
    2    2
    3    4    3
    5   14    9    5
    7   28   33   16    7
   11   69  104   74   29   11
   15  134  294  263  142   47   15

Links

Crossrefs

Row sums are A007716. First and last columns are both A000041.
Cf. A034691, A255397, A255903, A255906, A317532, A334550.

Programs

  • Mathematica
    permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
c[p_List, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
M[m_, n_, k_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
T[n_, k_] := M[k, n, n] - M[k - 1, n, n];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 08 2020, after Andrew Howroyd *)
  • PARI
    \\ See A318795 for definition of M.
T(n,k)={M(k, n, n) - M(k-1, n, n)}
for(n=1, 10, for(k=1, n, print1(T(n,k),", "));print) \\ Andrew Howroyd, Dec 28 2019
  • PARI
    \\ Faster version.
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, n)={1/prod(j=1, #q, (1-x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}
G(m,n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}
A(n,m=n)={my(p=sum(k=0, m, G(k,n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}
{ my(T=A(10)); for(n=1, #T, print(T[n,1..n])) } \\ Andrew Howroyd, Aug 30 2020

Extensions

Terms a(29) and beyond from Andrew Howroyd, Dec 28 2019

A381718 Number of normal multiset partitions of weight n into sets with distinct sums.

Original entry on oeis.org

1, 1, 2, 6, 23, 106, 549, 3184, 20353, 141615, 1063399, 8554800, 73281988, 665141182, 6369920854, 64133095134, 676690490875, 7462023572238, 85786458777923, 1025956348473929, 12739037494941490
Offset: 0

Views

Author

Gus Wiseman, Mar 26 2025

Keywords

Comments

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

Examples

			The a(1) = 1 through a(3) = 6 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,2}}
                    {{2},{1,3}}
                    {{1},{2},{3}}
The a(4) = 23 factorizations:
  2*3*6  5*30    3*30    2*30    210
         10*15   6*15    6*10    2*105
         2*5*15  2*3*15  2*3*10  3*70
         3*5*10                  5*42
                                 7*30
                                 6*35
                                 10*21
                                 2*3*35
                                 2*5*21
                                 2*7*15
                                 3*5*14
                                 2*3*5*7

Crossrefs

For distinct blocks instead of sums we have A116539, see A050326.
Without distinct sums we have A116540 (normal set multipartitions).
Twice-partitions of this type are counted by A279785.
Without strict blocks we have A326519.
Factorizations of this type are counted by A381633.
For constant instead of strict blocks we have A382203.
For distinct sizes instead of sums we have A382428, non-strict blocks A326517.
For equal instead of distinct block-sums we have A382429, non-strict blocks A326518.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A255903, A275780, A317532, A317583, A321469, A381635, A382204, A382214, A382216.

Programs

  • Mathematica
    allnorm[n_Integer]:=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[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

Extensions

a(10)-a(11) from Robert Price, Mar 31 2025
a(12)-a(20) from Christian Sievers, Apr 05 2025

A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 5, 4, 3, 15, 15, 11, 7, 5, 52, 52, 36, 21, 12, 7, 203, 203, 135, 74, 38, 19, 11, 877, 877, 566, 296, 141, 64, 30, 15, 4140, 4140, 2610, 1315, 592, 250, 105, 45, 22, 21147, 21147, 13082, 6393, 2752, 1098, 426, 165, 67, 30, 115975, 115975, 70631, 33645, 13960, 5317, 1940, 696, 254, 97, 42
Offset: 0

Views

Author

Alois P. Heinz, Jul 16 2021

Keywords

Comments

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

Examples

			A(2,2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.
Square array A(n,k) begins:
   1,  1,   2,    5,   15,    52,    203,     877,    4140, ...
   1,  2,   5,   15,   52,   203,    877,    4140,   21147, ...
   2,  4,  11,   36,  135,   566,   2610,   13082,   70631, ...
   3,  7,  21,   74,  296,  1315,   6393,   33645,  190085, ...
   5, 12,  38,  141,  592,  2752,  13960,   76464,  448603, ...
   7, 19,  64,  250, 1098,  5317,  28009,  158926,  963913, ...
  11, 30, 105,  426, 1940,  9722,  52902,  309546, 1933171, ...
  15, 45, 165,  696, 3281, 16972,  95129,  572402, 3670878, ...
  22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
  ...

Links

Crossrefs

Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
Main diagonal gives A346424.
Antidiagonal sums give A346428.
Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.

Programs

  • Maple
    s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
      combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
    end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..10);
  • Mathematica
    s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)

Formula

A(n,k) = A001055(A000079(n)*A070826(k+1)).
A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).
A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).

A317532 Regular triangle read by rows: T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 34, 26, 8, 16, 124, 168, 76, 16, 32, 448, 962, 674, 208, 32, 64, 1568, 5224, 5344, 2392, 544, 64, 128, 5448, 27336, 39834, 24578, 7816, 1376, 128, 256, 18768, 139712, 283864, 236192, 99832, 24048, 3392, 256, 512, 64448, 702496, 1960320, 2161602, 1186866, 370976, 70656, 8192, 512
Offset: 1

Views

Author

Gus Wiseman, Jul 30 2018

Keywords

Examples

			The T(3,2) = 8 multiset partitions:
  {{1},{1,1}}
  {{1},{2,2}}
  {{2},{1,2}}
  {{1},{1,2}}
  {{2},{1,1}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
Triangle begins:
    1
    2    2
    4    8    4
    8   34   26    8
   16  124  168   76   16
   32  448  962  674  208   32
  ...

Links

Crossrefs

Row sums are A255906.
Cf. A007716, A034691, A255397, A255903.

Programs

  • Mathematica
    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]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],Length[#]==k&]],{n,7},{k,n}]
  • PARI
    \\ here B(n,k) is A239473(n,k).
B(n,k)={sum(r=k, n, binomial(r, k)*(-1)^(r-k))}
Row(n)={Vecrev(sum(j=1, n, B(n,j)*polcoef(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^binomial(k+j-1,j-1)), n))/y)}
{ for(n=1, 10, print(Row(n))) } \\ Andrew Howroyd, Dec 31 2019

Extensions

Terms a(29) and beyond from Andrew Howroyd, Dec 31 2019

A317752 Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.

Original entry on oeis.org

0, 1, 8, 49, 305, 1984, 13686, 100124, 776885, 6386677, 55532358, 509549386, 4921352952, 49899820572, 529807799836, 5876162077537, 67928460444139, 816764249684450, 10195486840926032, 131896905499007474, 1765587483656124106, 24419774819813602870
Offset: 1

Views

Author

Gus Wiseman, Aug 06 2018

Keywords

Comments

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

Examples

			The a(3) = 8 multiset partitions with empty intersection:
  {{2},{1,1}}
  {{1},{2,2}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{1},{2}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Links

Crossrefs

Cf. A007716, A255903, A255906, A317073, A281116, A317077, A317532, A317533.
Cf. A317748, A317751, A317755, A317757, A317776.

Programs

  • Mathematica
    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]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,allnorm[n]}]],{n,6}]
  • PARI
    P(n,k)={1/prod(i=1, n, (1 - x^i*y + O(x*x^n))^binomial(k+i-1, k-1))}
R(n,k)={my(p=P(n,k), q=p/(1-y+O(y*y^n))); Vec(sum(i=2, n, polcoef(p,i,y) + polcoef(q,i,y)*sum(j=1, n\i, (-1)^j*binomial(k,j)*x^(i*j))), -n)}
seq(n)={sum(k=2, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Feb 05 2021

Extensions

Terms a(9) and beyond from Andrew Howroyd, Feb 05 2021

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 208, 434
Offset: 0

Views

Author

Gus Wiseman, Mar 29 2025

Keywords

Comments

We call a multiset normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

Examples

			The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
so is counted under a(5).
The multisets counted by A382214 but not by A382216 are:
  {1,1,1,1,2,2,3,3,3}
  {1,1,2,2,2,2,3,3,3}
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Crossrefs

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, A116539.
The complement is counted by A382202.
Without distinct sums we have A382214, complement A292432.
The case of a unique choice is counted by A382459, without distinct sums A382458.
For Heinz numbers: A293243, A381806, A382075, A382200.
For integer partitions: A381990, A381992, A382077, A382078.
Strong version: A382523, A382430, A381996, A292444.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.

Programs

  • Mathematica
    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[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]

A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 210, 436, 894
Offset: 0

Views

Author

Gus Wiseman, Mar 29 2025

Keywords

Comments

First differs from A382216 at a(9) = 210, A382216(9) = 208.
We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

Examples

			The normal multiset {1,1,1,1,2,2,3,3,3} has partition {{1},{3},{1,2},{1,3},{1,2,3}}, so is counted under a(9).
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Crossrefs

Factorizations of this type are counted by A050326, distinct sums A381633.
Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
The complement is counted by A292432.
Twice-partitions of this type are counted by A358914, distinct sums A279785.
The strong version is A381996, complement A292444.
For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
For distinct sums we have A382216, complement A382202.
The case of a unique choice is counted by A382458, distinct sums A382459.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A317532, A326517, A326519, A381990, A381992, A382428, A382460.

Programs

  • Mathematica
    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[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]

A075196 Table T(n,k) by antidiagonals: T(n,k) = number of partitions of n balls of k colors.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 14, 5, 5, 20, 38, 33, 7, 6, 30, 80, 117, 70, 11, 7, 42, 145, 305, 330, 149, 15, 8, 56, 238, 660, 1072, 906, 298, 22, 9, 72, 364, 1260, 2777, 3622, 2367, 591, 30, 10, 90, 528, 2198, 6174, 11160, 11676, 6027, 1132, 42, 11, 110, 735, 3582, 12292, 28784, 42805, 36450, 14873, 2139, 56
Offset: 1

Views

Author

Christian G. Bower, Sep 07 2002

Keywords

Comments

For k>=1, n->infinity is log(T(n,k)) ~ (1+1/k) * k^(1/(k+1)) * Zeta(k+1)^(1/(k+1)) * n^(k/(k+1)). - Vaclav Kotesovec, Mar 08 2015

Examples

			Square array T(n,k) begins:
  1,  2,   3,    4,    5, ...
  2,  6,  12,   20,   30, ...
  3, 14,  38,   80,  145, ...
  5, 33, 117,  305,  660, ...
  7, 70, 330, 1072, 2777, ...

Links

Crossrefs

Columns 1-10: A000041, A005380, A217093, A255050, A255052, A270239, A270240, A270241, A270242, A270243.
Rows 1-3: A000027, A002378, A162147.
Main diagonal: A075197.
Cf. A255903.

Programs

  • Maple
    with(numtheory):
A:= proc(n, k) option remember; local d, j;
      `if`(n=0, 1, add(add(d*binomial(d+k-1, k-1),
       d=divisors(j)) *A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 26 2012
  • Mathematica
    Transpose[Table[nn=6;p=Product[1/(1- x^i)^Binomial[i+n,n],{i,1,nn}];Drop[CoefficientList[Series[p,{x,0,nn}],x],1],{n,0,nn}]]//Grid  (* Geoffrey Critzer, Sep 27 2012 *)

Formula

T(n,k) = Sum_{i=0..k} C(k,i) * A255903(n,i). - Alois P. Heinz, Mar 10 2015

A327116 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 2, 15, 27, 15, 0, 3, 32, 102, 124, 52, 0, 4, 65, 319, 656, 600, 203, 0, 5, 124, 897, 2780, 4210, 3084, 877, 0, 6, 230, 2346, 10305, 23040, 27567, 16849, 4140, 0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147
Offset: 0

Views

Author

Alois P. Heinz, Sep 13 2019

Keywords

Examples

			T(3,2) = 6; 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   2;
  0, 2,   6,    5;
  0, 2,  15,   27,    15;
  0, 3,  32,  102,   124,     52;
  0, 4,  65,  319,   656,    600,    203;
  0, 5, 124,  897,  2780,   4210,   3084,    877;
  0, 6, 230, 2346, 10305,  23040,  27567,  16849,  4140;
  0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147;
  ...

Links

Crossrefs

Columns k=0-2 give: A000007, A000009 (for n>0), A327598.
Main diagonal gives A000110.
Row sums give A317776.
T(2n,n) gives A327556.
Cf. A255903, A326914, A326962, A327117, A327557, A309973.

Programs

  • Maple
    C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    c = Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A327557(n).
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