A229223 -id:A229223 - cp's OEIS frontend

A229226 The partition function G(n,9).

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678558, 4213452, 27642837, 190882290, 1382779413, 10478259030, 82844940414, 681863474058, 5830425411936, 51698581146426, 474582397380708, 4503425395487976, 44113612993755306, 445502134752984696

Offset: 0

Alois P. Heinz, Sep 16 2013

nonn

Number G(n,9) of set partitions of {1,...,n} into sets of size at most 9.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=9 of A229223.

Cf. A276929.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(n, 9):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n-1, i-1), i=1..min(n, 9)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2016

CoefficientList[Exp[Sum[x^j/j!, {j, 1, 9}]] + O[x]^25, x]*Range[0, 24]! (* Jean-François Alcover, May 21 2018 *)

E.g.f.: exp(Sum_{j=1..9} x^j/j!).

A229227 The partition function G(n,10).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678569, 4213584, 27644267, 190897305, 1382935569, 10479884654, 82861996310, 682044632178, 5832378929502, 51720008131148, 474821737584174, 4506150050048604, 44145239041717738, 445876518513670356

Offset: 0

Alois P. Heinz, Sep 16 2013

nonn

Number G(n,10) of set partitions of {1,...,n} into sets of size at most 10.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=10 of A229223.

Cf. A276930.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(n, 10):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n-1, i-1), i=1..min(n, 10)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2016

CoefficientList[Exp[Sum[x^j/j!, {j, 1, 10}]] + O[x]^25, x]*Range[0, 24]! (* Jean-François Alcover, May 21 2018 *)

E.g.f.: exp(Sum_{j=1..10} x^j/j!).

A229228 Number of set partitions of {1,...,2n} into sets of size at most n.

Original entry on oeis.org

1, 1, 10, 166, 3795, 112124, 4163743, 190168577, 10468226150, 681863474058, 51720008131148, 4506628734688128, 445956917001833090, 49631199898024188422, 6160538225093750695800, 846748983034696433927334, 128064669166890886264698699, 21195039362681903376709497444

Offset: 0

Alois P. Heinz, Sep 16 2013

nonn

			a(2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=2 of A229243.

Cf. A229223.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(2*n, n):
seq(a(n), n=0..20);

G[n_, k_] := G[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[G[n - k*j, k - 1]*n!/ k!^j/(n - k*j)!/j!, {j, 0, n/k}]]];
Table[G[2n, n], {n, 0, 20}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) = (2n)! * [x^(2n)] exp(Sum_{j=1..n} x^j/j!).

a(n) = A229223(2n,n).

A229229 Number of set partitions of {1,...,n^2} into sets of size at most n.

Original entry on oeis.org

1, 1, 10, 12644, 6631556521, 3282701194678476257, 3025262978042089315465899013351, 9292286146024114784457467780130028866860171013, 158655194198118596873150397161518177395553186289541468458000908304

Offset: 0

Alois P. Heinz, Sep 16 2013

nonn

			a(2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.

Alois P. Heinz, Table of n, a(n) for n = 0..24

Main diagonal of A229243.

Cf. A229223.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(n^2, n):
seq(a(n), n=0..10);

G[n_, k_] := G[n, k] = Module[{j, pc}, Which[k>n, G[n, n], n==0, 1, k<1, 0, True, pc = G[n-k, k]; For[j = k-1, j >= 1, j--, pc = pc*(n-j)/j + G[n-j, k]]; pc]]; a[n_] := G[n^2, n]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) = (n^2)! * [x^(n^2)] exp(Sum_{j=1..n} x^j/j!).

a(n) = A229223(n^2,n).

A227223 Number of set partitions of {1,...,2^n} into sets of size at most n.

Original entry on oeis.org

0, 1, 10, 2780, 6631556521, 71669271794142235712392433, 78417479379491793666843945562521255790293292270929676484784808001

Offset: 0

Alois P. Heinz, Sep 19 2013

nonn

			a(2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.

Alois P. Heinz, Table of n, a(n) for n = 0..9

Cf. A229223.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(2^n, n):
seq(a(n), n=0..7);

G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n-k, k]; For[j = k-1, j >= 1, j--, g = g*(n-j)/j + G[n-j, k]]; g]]; a[n_] := G[2^n, n]; Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)

a(n) = (2^n)! * [x^(2^n)] exp(Sum_{j=1..n} x^j/j!).

a(n) = A229223(2^n,n).

A229413 Number of set partitions of {1,...,3n} into sets of size at most n.

Original entry on oeis.org

1, 1, 76, 12644, 3305017, 1245131903, 654277037674, 467728049807348, 443694809361207824, 544852927413901502514, 846359710104516310431744, 1629392161877794034658847500, 3819592516111353522143561652540, 10738740219595085951726635839975852

Offset: 0

Alois P. Heinz, Sep 22 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=3 of A229243.

Cf. A229223.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(3*n, n):
seq(a(n), n=0..20);

G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n - k, k]; For[j = k - 1, j >= 1, j--, g = g(n-j)/j + G[n - j, k]]; g]];
a[n_] := G[3n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

a(n) = (3n)! * [x^(3n)] exp(Sum_{j=1..n} x^j/j!).

a(n) = A229223(3n,n).

A229414 Number of set partitions of {1,...,3n} into sets of size at most 3.

Original entry on oeis.org

1, 5, 166, 12644, 1680592, 341185496, 97620050080, 37286121988256, 18280749571449664, 11168256342434121152, 8306264068494786829696, 7380771881944947770497280, 7715405978050522488223499776, 9365880670184268387214967727104, 13058232187415887547449498864463872

Offset: 0

Alois P. Heinz, Sep 22 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Row n=3 of A229243.

a:= proc(n) option remember; `if`(n<3, [1, 5, 166][n+1],
      ((108*n^2-72*n+4)*a(n-1)-6*(n-1)*(3*n-5)*(27*n^2-48*n+10)*a(n-2)
       +9*(n-1)*(n-2)*(3*n-1)*(3*n-7)*(3*n-5)*(3*n-8)*a(n-3))/8)
    end:
seq(a(n), n=0..20);

G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n - k, k]; For[j = k - 1, j >= 1, j--, g = g(n-j)/j + G[n - j, k]]; g]];
a[n_] := G[3n, 3];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz in A229243 *)

a(n) = (3n)! * [x^(3n)] exp(x + x^2/2 + x^3/6).

a(n) = A001680(3n) = A229223(3n,3).

A379707 Number of nonempty labeled antichains of subsets of [n] such that all subsets except possibly those of the largest size are disjoint.

Original entry on oeis.org

1, 2, 5, 19, 133, 2605, 1128365, 68731541392, 1180735736455875189405, 170141183460507927984536600089529165335, 7237005577335553223087828975127304180898559033209149835788539833222132944557

Offset: 0

John Tyler Rascoe, Dec 30 2024

			For n < 4 all nonempty labeled antichains are counted. When n=6 antichains such as {{1,2,6},{1,4},{1,5}} are not counted, while {{1,2,4},{1,2,6},{3},{5}} is counted.

Cf. A000372, A014466, A229223, A245567, A305844, A379706.

from math import comb
def rS2(n,k,m):
    if n < 1 and k < 1: return 1
    elif n < 1 or k < 1: return 0
    else: return k*rS2(n-1,k,m) + rS2(n-1,k-1,m)- comb(n-1,m)*rS2(n-1-m,k-1,m)
def A229223(n,k):
    return sum(rS2(n,x,k) for x in range(n+1))
def A379707(n):
    return 1+sum(sum(comb(n,i)*(2**comb(n-i,s)-1)*A229223(i,s-1) for i in range(n-s+1)) for s in range(1,n+1))

a(n) = 1 + Sum_{s=1..n} (Sum_{i=0..n-s} binomial(n,i) * (2^binomial(n-i,s) - 1) * A229223(i,s-1)).

A295343 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j!).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, 0, -1, 0, 1, -1, 0, 2, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -1, 0, 1, 2, -6, 1, 0, 1, -1, 0, 1, 1, -6, 16, -1, 0, 1, -1, 0, 1, 1, -1, -14, 20, 1, 0, 1, -1, 0, 1, 1, -2, -14, 20, -132, -1, 0, 1, -1, 0, 1, 1, -2, -8, -15, 204, -28, 1, 0, 1, -1, 0, 1, 1, -2, -9, -15, 99, 28, 1216, -1, 0

Offset: 0

Ilya Gutkovskiy, Nov 20 2017

			Square array begins:
1,  1,  1,  1,  1,  1,  ...
0, -1  -1, -1, -1, -1,  ...
0,  1,  0,  0,  0,  0,  ...
0, -1,  2,  1,  1,  1,  ...
0,  1, -2,  2,  1,  1,  ...
0, -1, -6, -6, -1, -2,  ...

Columns k=0..3 give A000007, A033999, A001464, A014775.
Main diagonal gives A000587.
Cf. A229223.

Table[Function[k, n! SeriesCoefficient[Exp[-Sum[x^i/i!, {i, 1, k}]], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[1 - Exp[x] Gamma[k + 1, x]/Gamma[k + 1]], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(-Sum_{j=1..k} x^j/j!).

cp's OEIS Frontend

A229226 The partition function G(n,9).

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A229227 The partition function G(n,10).

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A229228 Number of set partitions of {1,...,2n} into sets of size at most n.

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A229229 Number of set partitions of {1,...,n^2} into sets of size at most n.

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A227223 Number of set partitions of {1,...,2^n} into sets of size at most n.

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A229413 Number of set partitions of {1,...,3n} into sets of size at most n.

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A229414 Number of set partitions of {1,...,3n} into sets of size at most 3.

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A379707 Number of nonempty labeled antichains of subsets of [n] such that all subsets except possibly those of the largest size are disjoint.

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