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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 12 results. Next

A271423 Number T(n,k) of set partitions of [n] with maximal block length multiplicity equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 5, 9, 0, 1, 0, 16, 25, 10, 0, 1, 0, 82, 70, 35, 15, 0, 1, 0, 169, 406, 245, 35, 21, 0, 1, 0, 541, 2093, 1036, 385, 56, 28, 0, 1, 0, 2272, 10935, 4984, 2331, 504, 84, 36, 0, 1, 0, 17966, 41961, 37990, 13335, 3717, 840, 120, 45, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Apr 07 2016

Keywords

Comments

At least one block length occurs exactly k times, and all block lengths occur at most k times.

Examples

			T(4,1) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
T(4,2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,4) = 1: 1|2|3|4.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     0,     1;
  0,     5,     9,     0,     1;
  0,    16,    25,    10,     0,    1;
  0,    82,    70,    35,    15,    0,   1;
  0,   169,   406,   245,    35,   21,   0,   1;
  0,   541,  2093,  1036,   385,   56,  28,   0,  1;
  0,  2272, 10935,  4984,  2331,  504,  84,  36,  0, 1;
  0, 17966, 41961, 37990, 13335, 3717, 840, 120, 45, 0, 1;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A007837 (for n>0), A271731, A271732, A271733, A271734, A271735, A271736, A271737, A271738, A271739.
Row sums give A000110.
Main diagonal gives A000012.
T(2n,n) gives A271425.
Cf. A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
    end:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]] * b[n - i*j, i - 1, k]/j!, {j, 0, Min[k, n/i]}]]]; T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *)

A271426 Number of set partitions of [n] with minimal block length multiplicity equal to one.

Original entry on oeis.org

0, 1, 1, 4, 11, 51, 132, 771, 3089, 18388, 96423, 627529, 3349018, 24510305, 155908651, 1171494200, 8647906143, 71603237483, 572103586280, 5172888505403, 43344865682187, 416735802793600, 3830340992280773, 38239507035358011, 374336654847685014
Offset: 0

Views

Author

Alois P. Heinz, Apr 07 2016

Keywords

Comments

At least one block length occurs exactly once.

Examples

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.

Links

Crossrefs

Column k=1 of A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0,$k..n/i})))
    end:
a:= n-> b(n$2, 1)-b(n$2, 2):
seq(a(n), n=0..30);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 1] - b[n, n, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 07 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,1).

A271715 Number of set partitions of [3n] with minimal block length multiplicity equal to n.

Original entry on oeis.org

1, 4, 55, 1540, 67375, 4239235, 383563180, 51925673800, 10652498631775, 3139051466175625, 1228555090548911125, 602267334323068414000, 357161594247065690582500, 250870551734754490461422500, 205672479804595549379158525000, 194557626586812183102927448930000
Offset: 0

Views

Author

Alois P. Heinz, Apr 12 2016

Keywords

Links

Crossrefs

Cf. A271424, A372722.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<5,
      [1, 4, 55, 1540, 67375][n+1], ((2*(3*n-2))*
       (3*n-1)*(n^2-n-9)*a(n-1) -(3*(n-3))*(3*n-1)*
       (3*n-4)*(3*n-2)*(3*n-5)*a(n-2))/(4*n*(n-4)))
    end:
seq(a(n), n=0..20);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i&, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]] // Union}]]];
a[n_] := If[n==0, 1, b[3n, 3n, n] - b[3n, 3n, n+1]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz in A271424 *)

Formula

a(n) = A271424(3n,n).
Recursion: see Maple program.
For n>0, a(n) = (3^n + n!)*(3*n)! / (6^n * (n!)^2). - Vaclav Kotesovec, Apr 16 2016
a(n) = A372722(3n,n). - Alois P. Heinz, May 11 2024

A271762 Number of set partitions of [n] with minimal block length multiplicity equal to two.

Original entry on oeis.org

1, 0, 3, 0, 55, 105, 945, 1218, 15456, 26785, 705573, 2502786, 32988670, 169561483, 1757881723, 10231748010, 84389906941, 540218433147, 6899156019034, 41756989590256, 554960234199955, 4793361957432730, 59690079139252499, 558283841454550850, 7093218105977514525
Offset: 2

Views

Author

Alois P. Heinz, Apr 13 2016

Keywords

Examples

			a(4) = 3: 12|34, 13|24, 14|23.

Links

Crossrefs

Column k=2 of A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
    end:
a:= n-> b(n$2, 2)-b(n$2, 3):
seq(a(n), n=2..30);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 2] - b[n, n, 3];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,2).

A271763 Number of set partitions of [n] with minimal block length multiplicity equal to three.

Original entry on oeis.org

1, 0, 0, 15, 0, 0, 1540, 3150, 24255, 81235, 496210, 605605, 36987951, 13833820, 849333940, 24419945732, 111237098546, 1219799147204, 16146398449224, 109697049177254, 1037441240056529, 9042707959752775, 84237798887033660, 614681985047225810
Offset: 3

Views

Author

Alois P. Heinz, Apr 13 2016

Keywords

Examples

			a(6) = 15: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 13|25|46, 13|26|45, 14|23|56, 15|23|46, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.

Links

Crossrefs

Column k=3 of A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
    end:
a:= n-> b(n$2, 3)-b(n$2, 4):
seq(a(n), n=3..30);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 3] - b[n, n, 4];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,3).

A271764 Number of set partitions of [n] with minimal block length multiplicity equal to four.

Original entry on oeis.org

1, 0, 0, 0, 105, 0, 0, 0, 67375, 135135, 1261260, 675675, 50925875, 97847750, 703993290, 6215737710, 228687298476, 58017429575, 11262925616250, 72813288304295, 2841531210935725, 11311740884766630, 252469888906590355, 2207276997956560530, 28579415631325499655
Offset: 4

Views

Author

Alois P. Heinz, Apr 13 2016

Keywords

Links

Crossrefs

Column k=4 of A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
    end:
a:= n-> b(n$2, 4)-b(n$2, 5):
seq(a(n), n=4..30);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 4] - b[n, n, 5];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,4).

A271765 Number of set partitions of [n] with minimal block length multiplicity equal to five.

Original entry on oeis.org

1, 0, 0, 0, 0, 945, 0, 0, 0, 0, 4239235, 7567560, 82702620, 41351310, 1658646990, 24448068645, 117626817945, 239611442070, 8260908743395, 1834189492520, 4508736346382576, 2979073800027325, 256635727575051825, 2371542394294648575, 16374593589666387075
Offset: 5

Views

Author

Alois P. Heinz, Apr 13 2016

Keywords

Links

Crossrefs

Column k=5 of A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
    end:
a:= n-> b(n$2, 5)-b(n$2, 6):
seq(a(n), n=5..30);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 5] - b[n, n, 6];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,5).

A271766 Number of set partitions of [n] with minimal block length multiplicity equal to six.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 10395, 0, 0, 0, 0, 0, 383563180, 523783260, 6547290750, 3055402350, 157964301495, 14054850810, 34828180582195, 91670862398500, 448593283888750, 11612610774464700, 7681370284312725, 6594450798260325, 179804372693675480751, 11896760875264765500
Offset: 6

Views

Author

Alois P. Heinz, Apr 13 2016

Keywords

Links

Crossrefs

Column k=6 of A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
    end:
a:= n-> b(n$2, 6)-b(n$2, 7):
seq(a(n), n=6..30);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 6] - b[n, n, 7];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,6).

A271767 Number of set partitions of [n] with minimal block length multiplicity equal to seven.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 135135, 0, 0, 0, 0, 0, 0, 51925673800, 43212118950, 607370338575, 265034329560, 17166996346500, 1305093289500, 584129638842750, 56071685084790375, 176898040019801100, 518112685551586125, 26529011711988035250, 4672320885518286000
Offset: 7

Views

Author

Alois P. Heinz, Apr 13 2016

Keywords

Links

Crossrefs

Column k=7 of A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
    end:
a:= n-> b(n$2, 7)-b(n$2, 8):
seq(a(n), n=7..35);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 7] - b[n, n, 8];
Table[a[n], {n, 7, 35}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,7):

A271768 Number of set partitions of [n] with minimal block length multiplicity equal to eight.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 2027025, 0, 0, 0, 0, 0, 0, 0, 10652498631775, 4141161399375, 64602117830250, 26428139112375, 2096632369581750, 137561852302875, 80768458994973750, 609202488769875, 158980016052580597875, 353341814230502847750, 1344898884799733513250
Offset: 8

Views

Author

Alois P. Heinz, Apr 13 2016

Keywords

Links

Crossrefs

Column k=8 of A271424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
    end:
a:= n-> b(n$2, 8)-b(n$2, 9):
seq(a(n), n=8..35);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 8] - b[n, n, 9];
Table[a[n], {n, 8, 35}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,8).
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