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 1-7 of 7 results.

A343664 Number of partitions of an n-set without blocks of size 4.

Original entry on oeis.org

1, 1, 2, 5, 14, 47, 173, 702, 3125, 14910, 76495, 418035, 2418397, 14791597, 95093612, 641094695, 4521228732, 33250447919, 254585084539, 2024995604762, 16702070759557, 142642458681486, 1259387604241013, 11479967000116911, 107910143688962037, 1044735841257587203, 10407104137208385924
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Links

Crossrefs

Cf. A000110, A000296, A027338, A097514, A124504, A343665, A343666, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=4, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..26);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 26; CoefficientList[Series[Exp[Exp[x] - 1 - x^4/4!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 4 k]/((n - 4 k)! k! (4!)^k), {k, 0, Floor[n/4]}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 4, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

Formula

E.g.f.: exp(exp(x) - 1 - x^4/4!).
a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * Bell(n-4*k) / ((n-4*k)! * k! * (4!)^k).

A343665 Number of partitions of an n-set without blocks of size 5.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 197, 835, 3860, 19257, 102997, 586170, 3535645, 22496437, 150454918, 1054235150, 7718958995, 58905868192, 467530598983, 3851775136517, 32881385742460, 290387471713872, 2649226725182823, 24934118754400767, 241809265181914545, 2413608066257526577
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027339, A097514, A124504, A343664, A343666, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=5, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 5 k]/((n - 5 k)! k! (5!)^k), {k, 0, Floor[n/5]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 5, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^5/5!).
a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * Bell(n-5*k) / ((n-5*k)! * k! * (5!)^k).

A343666 Number of partitions of an n-set without blocks of size 6.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 870, 4084, 20727, 112825, 654546, 4026487, 26145511, 178550986, 1278168860, 9564026947, 74615547996, 605593775899, 5103054929621, 44564754448972, 402677613100491, 3759094788129312, 36205919126040190, 359340174509911325, 3670825700549853053
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027340, A097514, A124504, A343664, A343665, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=6, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^6/6!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 6 k]/((n - 6 k)! k! (6!)^k), {k, 0, Floor[n/6]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 6, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^6/6!).
a(n) = n! * Sum_{k=0..floor(n/6)} (-1)^k * Bell(n-6*k) / ((n-6*k)! * k! * (6!)^k).

A343667 Number of partitions of an n-set without blocks of size 7.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 876, 4132, 21075, 115375, 673620, 4172413, 27296089, 187891174, 1356343385, 10238632307, 80615222404, 660560758879, 5621465069117, 49594663447612, 452846969975391, 4273130715906123, 41612346388251187, 417668648929556073, 4315893703814296053
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027341, A097514, A124504, A343664, A343665, A343666, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=7, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^7/7!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 7 k]/((n - 7 k)! k! (7!)^k), {k, 0, Floor[n/7]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 7, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^7/7!).
a(n) = n! * Sum_{k=0..floor(n/7)} (-1)^k * Bell(n-7*k) / ((n-7*k)! * k! * (7!)^k).

A343669 Number of partitions of an n-set without blocks of size 9.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115965, 678460, 4212497, 27633712, 190795218, 1381942530, 10470109267, 82764226404, 681048663329, 5822029128397, 51610194855972, 473631475252041, 4492967510009533, 43996047374513046, 444151309687221889, 4617189912288741028
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027343, A097514, A124504, A343664, A343665, A343666, A343667, A343668.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=9, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^9/9!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 9 k]/((n - 9 k)! k! (9!)^k), {k, 0, Floor[n/9]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 9, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^9/9!).
a(n) = n! * Sum_{k=0..floor(n/9)} (-1)^k * Bell(n-9*k) / ((n-9*k)! * k! * (9!)^k).

A343791 Number of ordered partitions of an n-set without blocks of size 8.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545834, 7087243, 102247203, 1622625313, 28091415135, 526854986737, 10641264928479, 230281282588513, 5315605563021465, 130369438065006551, 3385496924633886429, 92800464391224494215, 2677652842774247060805, 81123688691904430522831
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 29 2021

Keywords

Crossrefs

Cf. A000670, A032032, A337058, A337059, A343668, A343787, A343788, A343789, A343790, A343792, A343793.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=8, 0, a(n-j)*binomial(n, j)), j=1..n))
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 29 2021
  • Mathematica
    nmax = 21; CoefficientList[Series[1/(2 + x^8/8! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 8, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

Formula

E.g.f.: 1 / (2 + x^8/8! - exp(x)).

A343671 Number of partitions of an n-set without blocks of size 10.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678559, 4213465, 27643007, 190884307, 1382802389, 10478516523, 82847813908, 681895648039, 5830788687491, 51702731250650, 474630475600569, 4503991075480297, 44120379612630694, 445584481578266277, 4634070027874688433
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027344, A097514, A124504, A343664, A343665, A343666, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
       j=10, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2023
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^10/10!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 10 k]/((n - 10 k)! k! (10!)^k), {k, 0, Floor[n/10]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 10, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^10/10!).
a(n) = n! * Sum_{k=0..floor(n/10)} (-1)^k * Bell(n-10*k) / ((n-10*k)! * k! * (10!)^k).
Showing 1-7 of 7 results.

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