A242896 -id:A242896 - cp's OEIS frontend

A242882 Number of compositions of n into parts with distinct multiplicities.

1, 1, 2, 2, 6, 12, 16, 40, 60, 82, 216, 538, 788, 2034, 3740, 6320, 13336, 27498, 42936, 93534, 173520, 351374, 734650, 1592952, 3033194, 6310640, 12506972, 25296110, 49709476, 101546612, 195037028, 391548336, 764947954, 1527004522, 2953533640, 5946359758

Offset: 0

Alois P. Heinz, May 25 2014

nonn

			a(0) = 1: the empty composition.
a(1) = 1: [1].
a(2) = 2: [1,1], [2].
a(3) = 2: [1,1,1], [3].
a(4) = 6: [1,1,1,1], [1,1,2], [1,2,1], [2,1,1], [2,2], [4].
a(5) = 12: [1,1,1,1,1], [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1], [5].

Vaclav Kotesovec, Table of n, a(n) for n = 0..264 (terms 0..200 from Alois P. Heinz)
Vaclav Kotesovec, What is the limit a(n)/2^n ?

Row sums of A242887 and of A242896.

Cf. A098859 (the same for partitions).

b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
      `if`(i<1, 0, add(`if`(j>0 and j in s, 0,
      b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i)))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..45);

b[n_, i_, s_] := b[n, i, s] = If[n == 0, Sum[j, {j, s}]!, If[i < 1, 0, Sum[If[j > 0 && MemberQ[s, j], 0, b[n - i*j, i - 1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, {}];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

a(n)={((r,k,b,w)->if(!k||!r, if(r,0,w!), sum(m=0, r\k, if(!m || !bittest(b,m), self()(r-k*m, k-1, bitor(b,1<Andrew Howroyd, Aug 31 2019

A242887 Number T(n,k) of compositions of n into parts with distinct multiplicities and with exactly k parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 3, 1, 0, 1, 0, 6, 4, 1, 0, 1, 1, 4, 4, 5, 1, 0, 1, 0, 9, 8, 15, 6, 1, 0, 1, 1, 9, 5, 15, 21, 7, 1, 0, 1, 0, 10, 8, 20, 6, 28, 8, 1, 0, 1, 1, 12, 12, 6, 96, 42, 36, 9, 1, 0, 1, 0, 15, 12, 30, 192, 168, 64, 45, 10, 1, 0, 1, 1, 13, 9, 20, 142, 238, 204, 93, 55, 11, 1

Offset: 0

Alois P. Heinz, May 25 2014

			T(5,1) = 1: [5].
T(5,3) = 6: [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1].
T(5,4) = 4: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1].
T(5,5) = 1: [1,1,1,1,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 0,  1;
  0, 1, 1,  3,  1;
  0, 1, 0,  6,  4,  1;
  0, 1, 1,  4,  4,  5,  1;
  0, 1, 0,  9,  8, 15,  6,  1;
  0, 1, 1,  9,  5, 15, 21,  7,  1;
  0, 1, 0, 10,  8, 20,  6, 28,  8, 1;
  0, 1, 1, 12, 12,  6, 96, 42, 36, 9, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A057427, A059841 (for n>1), A321773, A321774, A321775, A321776, A321777, A321778, A321779, A321780.
Row sums give A242882.
T(2n,n) gives A321772.
Cf. A242896.

b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
      `if`(i<1, 0, expand(add(`if`(j>0 and j in s, 0, x^j*
       b(n-i*j, i-1,`if`(j=0, s, s union {j}))/j!), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2, {})):
seq(T(n), n=0..16);

b[n_, i_, s_] := b[n, i, s] = If[n==0, Total[s]!, If[i<1, 0, Expand[Sum[ If[j>0 && MemberQ[s, j], 0, x^j*b[n-i*j, i-1, If[j==0, s, s ~Union~ {j}] ]/j!], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, {}]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

T(n)={Vecrev(((r,k,b,w)->if(!k||!r, if(r,0,w!*x^w), sum(m=0, r\k, if(!m || !bittest(b,m), self()(r-k*m, k-1, bitor(b,1<Andrew Howroyd, Aug 31 2019

A182485 Number of partitions of n into exactly k different parts with distinct multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 0, 3, 1, 0, 2, 3, 0, 4, 3, 0, 2, 8, 0, 4, 9, 0, 3, 12, 0, 4, 16, 1, 0, 2, 22, 4, 0, 6, 20, 5, 0, 2, 31, 12, 0, 4, 35, 16, 0, 4, 34, 24, 0, 5, 44, 33, 0, 2, 51, 52, 0, 6, 53, 57, 0, 2, 62, 89, 0, 6, 65, 100, 1, 0, 4, 68, 131, 5, 0, 4, 87

Offset: 0

Alois P. Heinz, May 01 2012

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,1], [2].
T(4,1) = 3: [1,1,1,1], [2,2], [4].
T(4,2) = 1: [2,1,1]; part 2 occurs once and part 1 occurs twice.
T(5,2) = 3: [2,1,1,1], [2,2,1], [3,1,1].
T(7,2) = 8: [2,1,1,1,1,1], [2,2,1,1,1], [2,2,2,1], [3,1,1,1,1], [3,2,2], [3,3,1], [4,1,1,1], [5,1,1].
T(10,1) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
T(10,3) = 1: [3,2,2,1,1,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 2;
  0, 2;
  0, 3,  1;
  0, 2,  3;
  0, 4,  3;
  0, 2,  8;
  0, 4,  9;
  0, 3, 12;
  0, 4, 16, 1;

Alois P. Heinz, Rows n = 0..120, flattened

Columns k=0-7 give: A000007, A000005, A182473, A212114, A212115, A212116, A212117, A212118.
Row sums give: A098859.
First row with length (t+1): A000292(t).
Cf. A242896 (the same for compositions):

b:= proc(n, i, t, s) option remember;
      `if`(nops(s)>t, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t, s)+
      add(`if`(j in s, 0, b(n-i*j, i-1, t, s union {j})), j=1..n/i))))
    end:
g:= proc(n) local i; for i while i*(i+1)*(i+2)/6<=n do od; i-1 end:
T:= n-> seq(b(n, n, k, {}) -b(n, n, k-1, {}), k=0..g(n)):
seq(T(n), n=0..30);

b[n_, i_, t_, s_] := b[n, i, t, s] = If[Length[s] > t, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, t, s] + Sum[If[MemberQ[s, j], 0, b[n-i*j, i-1, t, s ~Union~ {j}]], {j, 1, n/i}]]]]; g[n_] := Module[{i}, For[ i = 1, i*(i+1)*(i+2)/6 <= n , i++]; i-1 ]; t[n_] := Table [b[n, n, k, {}] - b[n, n, k-1, {}], {k, 0, g[n]}]; Table [t[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)

A242900 Number of compositions of n into exactly two different parts with distinct multiplicities.

Original entry on oeis.org

3, 10, 12, 38, 56, 79, 152, 251, 284, 594, 920, 1108, 2136, 3402, 4407, 8350, 12863, 17328, 33218, 52527, 70074, 133247, 214551, 294299, 547360, 883572, 1234509, 2284840, 3667144, 5219161, 9551081, 15386201, 22079741, 40061664, 64666975, 93985744, 168363731

Offset: 4

Alois P. Heinz, May 25 2014

nonn

			a(4) = 3: [2,1,1], [1,2,1], [1,1,2].
a(5) = 10: [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [2,2,1], [2,1,2], [1,2,2], [3,1,1], [1,3,1], [1,1,3].

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=2 of A242896.

Cf. A182473 (the same for partitions), A131661 (multiplicities may be equal).

with(numtheory):
a:= n-> add(add(add(`if`(dm, binomial((n-p*m)
        /d+m, m), 0), d=divisors(n-p*m)), m=1..n/p), p=2..n-1):
seq(a(n), n=4..60);

div[0] = {}; div[n_] := Divisors[n]; a[n_] := Sum[Sum[Sum[If[dJean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, Aug 21 2014

A246230 Number of compositions of n into exactly three different parts with distinct multiplicities.

Original entry on oeis.org

60, 285, 498, 1438, 2816, 5208, 11195, 24094, 38523, 85182, 148051, 255922, 512428, 991423, 1573152, 3318750, 5820718, 9712145, 19523028, 36637218, 58805129, 121712569, 216006156, 360123456, 720094287, 1340734558, 2184432420, 4453145090, 7987036106, 13417450294

Offset: 10

Alois P. Heinz, Aug 19 2014

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 10..220 (first 120 terms from Alois P. Heinz)

Column k=3 of A242896.

b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i<1, 0, Expand[ Sum[ If[j>0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n-i*j, i-1, If[j == 0, s, s ~Union~{j}]]/j!], {j, 0, n/i}]]]]; a[n_] := Coefficient[b[n, n, {}], x, 3]; Table[a[n], {n, 10, 40}] (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

A246231 Number of compositions of n into exactly four different parts with distinct multiplicities.

Original entry on oeis.org

12600, 78120, 189000, 549000, 1389960, 2858640, 6471699, 15289662, 29639082, 64025820, 134997382, 267550925, 545274648, 1161661901, 2223888268, 4552401195, 9185643748, 17829746962, 35102264675, 71613947350, 135793314152, 270167493481, 534784841445

Offset: 20

Alois P. Heinz, Aug 19 2014

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 20..220 (first 120 terms from Alois P. Heinz)

Column k=4 of A242896.

b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i<1, 0, Expand[ Sum[ If[j>0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n-i*j, i-1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]]]; a[n_] := Coefficient[b[n, n, {}], x, 4]; Table[a[n], {n, 20, 50}] (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

A246232 Number of compositions of n into exactly five different parts with distinct multiplicities.

Original entry on oeis.org

37837800, 290089800, 913512600, 2870988120, 8647198560, 20798177400, 52611741000, 134182896120, 299814913440, 688027838400, 1598171790600, 3415039782840, 7488313184520, 16753428299160, 34973088459120, 74787505653264, 162128661286152, 333915200009352

Offset: 35

Alois P. Heinz, Aug 19 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 35..120

Column k=5 of A242896.

b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i<1, 0, Expand[ Sum[ If[j>0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n-i*j, i-1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]]]; a[n_] := Coefficient[b[n, n, {}], x, 5]; Table[a[n], {n, 35, 60}] (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

A246233 Number of compositions of n into exactly six different parts with distinct multiplicities.

Original entry on oeis.org

2053230379200, 18772392038400, 73402986056400, 257607082933200, 886531102657200, 2509031227903200, 7078865111398080, 19909183290796800, 51039390957675120, 128693357256063600, 326570236365292920, 777083694958410840, 1847217791018559960, 4410908197048833480

Offset: 56

Alois P. Heinz, Aug 19 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 56..120

Column k=6 of A242896.

$RecursionLimit = 1000; b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i < 1, 0, Expand[ Sum[ If[j > 0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n - i*j, i - 1, If[j == 0, s, s~Union~{j}]]/j!], {j, 0, n/i}]]]]; a[n_] := Coefficient[b[n, n, {}], x, 6]; Table[a[n], {n, 56, 80}] (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

A246234 Number of compositions of n into exactly seven different parts with distinct multiplicities.

Original entry on oeis.org

2431106898187968000, 25830510793247160000, 121150160426367072000, 478913588068635720000, 1848522650426852400000, 6067325851827648408000, 19158973516499432616000, 59882095876893754464000, 173801554657753702680000, 487295849448575903736000

Offset: 84

Alois P. Heinz, Aug 19 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 84..120

Column k=7 of A242896.

$RecursionLimit = 1000; b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i < 1, 0, Expand[ Sum[ If[j > 0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n - i*j, i - 1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]]]; a[n_] := Coefficient[b[n, n, {}], x, 7]; Table[a[n], {n, 84, 100}] (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

cp's OEIS Frontend

A242882 Number of compositions of n into parts with distinct multiplicities.

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A242887 Number T(n,k) of compositions of n into parts with distinct multiplicities and with exactly k parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A182485 Number of partitions of n into exactly k different parts with distinct multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.

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A242900 Number of compositions of n into exactly two different parts with distinct multiplicities.

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A246230 Number of compositions of n into exactly three different parts with distinct multiplicities.

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A246231 Number of compositions of n into exactly four different parts with distinct multiplicities.

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A246232 Number of compositions of n into exactly five different parts with distinct multiplicities.

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A246233 Number of compositions of n into exactly six different parts with distinct multiplicities.

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A246234 Number of compositions of n into exactly seven different parts with distinct multiplicities.

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