A332747 -id:A332747 - cp's OEIS frontend

A332721 Number of compositions of n^2 with parts <= n, such that each element of [n] is used at least once.

1, 1, 3, 72, 5752, 1501620, 1171326960, 2571831080160, 15245263511750160, 236246829658682027760, 9325247205993698149853760, 917699267902161951609308035200, 221117091698491444413008381486903040, 128433050637127079872089064922773889126400

Offset: 0

Alois P. Heinz, Feb 21 2020

nonn

			a(2) = 3: 112, 121, 211.
a(3) = 72: 111123, 111132, 111213, 111231, 111312, 111321, 112113, 112131, 112311, 113112, 113121, 113211, 121113, 121131, 121311, 123111, 131112, 131121, 131211, 132111, 211113, 211131, 211311, 213111, 231111, 311112, 311121, 311211, 312111, 321111, 11223, 11232, 11322, 12123, 12132, 12213, 12231, 12312, 12321, 13122, 13212, 13221, 21123, 21132, 21213, 21231, 21312, 21321, 22113, 22131, 22311, 23112, 23121, 23211, 31122, 31212, 31221, 32112, 32121, 32211, 1233, 1323, 1332, 2133, 2313, 2331, 3123, 3132, 3213, 3231, 3312, 3321.

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A000079, A000290, A103488, A332716, A332747, A332796, A373118.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1, (p+n)!
       /(n+1)!, add(b(n-i*j, i-1, p+j)/(j+1)!, j=0..n/i))
    end:
a:= n-> b(n*(n-1)/2, n$2):
seq(a(n), n=0..15);

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p + n)!/(n + 1)!, Sum[b[n - i*j, i - 1, p + j]/(j + 1)!, {j, 0, n/i}]];
a[n_] := b[n(n - 1)/2, n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)

a(n) = A373118(n^2,n). - Alois P. Heinz, May 26 2024

A332796 Number of compositions of n^2 into parts >= n.

Original entry on oeis.org

1, 1, 2, 6, 26, 140, 882, 6349, 51284, 457704, 4459940, 47019819, 532485538, 6438774524, 82710138994, 1123798871990, 16090426592488, 241979954659728, 3811335657375786, 62712512310820402, 1075527196672980525, 19186234784992217621, 355349469934379290700

Offset: 0

Alois P. Heinz, Feb 24 2020

nonn

			a(0) = 1: (), the empty composition.
a(1) = 1: 1.
a(2) = 2: 22, 4.
a(3) = 6: 333, 36, 63, 45, 54, 9.
a(4) = 26: 4444, 556, 565, 655, 466, 646, 664, 457, 475, 547, 574, 745, 754, 448, 484, 844, 88, 79, 97, 6(10), (10)6, 5(11), (11)5, 4(12), (12)4, (16).

Alois P. Heinz, Table of n, a(n) for n = 0..170

Cf. A011782, A103488, A332716, A332721, A332747.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k), j=k..n))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..23);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k], {j, k, n}]];
a[n_] := b[n^2, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)

A333048 Number of compositions of n^2 into powers of n.

Original entry on oeis.org

1, 1, 6, 20, 96, 572, 3971, 31201, 272334, 2605268, 27042522, 302171806, 3611295430, 45911641817, 618074912240, 8776287336812, 130994094465946, 2049114914257540, 33504826964461451, 571285301051283841, 10136481840545237652, 186803012671904648805

Offset: 0

Alois P. Heinz, Mar 06 2020

nonn

			a(0) = 1: the empty composition.
a(1) = 1: 1.
a(2) = 6: 1111, 112, 121, 211, 22, 4.
a(3) = 20: 111111111, 1111113, 1111131, 1111311, 1113111, 1131111, 1311111, 3111111, 11133, 11313, 11331, 13113, 13131, 13311, 31113, 31131, 31311, 33111, 333, 9.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000290, A011782, A103488, A332721, A332747, A332796.

a:= n-> `if`(n<2, 1, 1+add(binomial(n*(n-j)+j, j), j=0..n)):
seq(a(n), n=0..21);

a(n) = 1 + Sum_{j=0..n} binomial(n*(n-j)+j,j) if n>1, a(0) = a(1) = 1.

cp's OEIS Frontend

A332721 Number of compositions of n^2 with parts <= n, such that each element of [n] is used at least once.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A332796 Number of compositions of n^2 into parts >= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A333048 Number of compositions of n^2 into powers of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula