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.

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A178979 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the shortest block has length k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 4, 0, 1, 11, 3, 0, 1, 41, 10, 0, 0, 1, 162, 30, 10, 0, 0, 1, 715, 126, 35, 0, 0, 0, 1, 3425, 623, 56, 35, 0, 0, 0, 1, 17722, 2934, 364, 126, 0, 0, 0, 0, 1, 98253, 15165, 2220, 210, 126, 0, 0, 0, 0, 1, 580317, 86900, 10560, 330, 462, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Geoffrey Critzer, Jan 02 2011

Keywords

Comments

Row sums are Bell numbers A000110.
Column 1 is A000296 (shifted).
From Peter Luschny, Apr 05 2011: (Start)
Sum_{k>1} T(n,k) = A000296(n) count the set partitions with blocks of size > 1.
T(n,1) = A000296(n-1) count the set partitions with blocks of size = 1. Thus for the Bell numbers A000110(n) = Sum_{k>=1} T(n,k) = A000296(n-1) + A000296(n). (End)

Examples

			T(4,2) = card ({12|34, 13|24, 14|23}) = 3. - _Peter Luschny_, Apr 05 2011
Triangle begins:
    1;
    1,   1;
    4,   0,  1;
   11,   3,  0,  1;
   41,  10,  0,  0,  1;
  162,  30, 10,  0,  0,  1;
  715, 126, 35,  0,  0,  0,  1;
  ...

Links

Crossrefs

Cf. A080510, A145877, A000110, A000296.

Programs

  • Maple
    g := k-> exp(x)*(1-(GAMMA(k,x)/GAMMA(k))); egf := k-> exp(g(k))-exp(g(k+1));
T := (n,k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..9); # Peter Luschny, Apr 05 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
       add(b(n-i*j, i+1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
T:= (n, k)-> b(n, k) -b(n, k+1):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 25 2016
  • Mathematica
    a[k_]:= Exp[x]-Sum[x^i/i!,{i,0,k}]; Transpose[Table[Range[20]! Rest[CoefficientList[Series[Exp[a[k-1]]-Exp[a[k]],{x,0,20}],x]],{k,1,9}]]//Grid

Formula

E.g.f. for column k: exp((exp(x) - Sum_{i=0..k-1} x^i/i!)) - exp((exp(x) - Sum_{i=0..k} x^i/i!)).
From Ludovic Schwob, Jan 15 2022: (Start)
T(2n,n) = A001700(n) = C(2n-1,n) for n>0.
T(2n-1,n-1) = A001700(n) = C(2n-1,n) for n>1. (End)

A229245 Number of set partitions of {1,...,n} with largest set of size 3.

Original entry on oeis.org

1, 4, 20, 90, 420, 2016, 10024, 51640, 276980, 1540440, 8899176, 53313624, 330835960, 2124646720, 14102514560, 96622736256, 682608577104, 4966188238080, 37166169295360, 285813960789280, 2256147419689856, 18263257380872064, 151466260791609600
Offset: 3

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Links

Crossrefs

Column k=3 of A080510.

Programs

  • Maple
    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,3)-G(n,2):
seq(a(n), n=3..30);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n - i j, i - 1] n!/ i!^j/(n - i j)!/j!, {j, 0, n/i}]]];
a[n_] := b[n, 3] - b[n, 2];
a /@ Range[3, 30] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz in A080510 *)

Formula

E.g.f.: exp(Sum_{j=1..3} x^j/j!) - exp(Sum_{j=1..2} x^j/j!).

A229246 Number of set partitions of {1,...,n} with largest set of size 4.

Original entry on oeis.org

1, 5, 30, 175, 1015, 6111, 38010, 244035, 1624425, 11187605, 79695616, 586787565, 4460703065, 34979737625, 282686608170, 2352035472141, 20130594430095, 177076978131795, 1599554765785900, 14826174374092235, 140904888153964011, 1372081103132069275
Offset: 4

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Links

Crossrefs

Column k=4 of A080510.

Programs

  • Maple
    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,4)-G(n,3):
seq(a(n), n=4..30);
  • Mathematica
    nn=25;Drop[Range[0,nn]!CoefficientList[Series[(Exp[x^4/4!]-1)Exp[x+x^2/2!+x^3/3!],{x,0,nn}],x],4] (* Geoffrey Critzer, Oct 09 2013 *)

Formula

E.g.f.: exp(Sum_{j=1..4} x^j/j!) - exp(Sum_{j=1..3} x^j/j!).

A229247 Number of set partitions of {1,...,n} with largest set of size 5.

Original entry on oeis.org

1, 6, 42, 280, 1890, 12978, 91938, 671616, 5064345, 39439400, 317158842, 2631497232, 22512271964, 198412838820, 1800062132940, 16795556650200, 161038724157045, 1585408383273330, 16013462706719170, 165819496710741720, 1759058150311036806, 19103856738729254206
Offset: 5

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Links

Crossrefs

Column k=5 of A080510.

Programs

  • Maple
    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,5)-G(n,4):
seq(a(n), n=5..30);
  • Mathematica
    nmin = size = 5; nmax = 30;
g[k_] := Exp[Sum[x^j/j!, {j, 1, k}]];
cc = CoefficientList[g[size]-g[size-1]+O[x]^(nmax+1), x]*Range[0, nmax]!;
a[n_] := cc[[n+1]];
a /@ Range[nmin, nmax] (* Jean-François Alcover, Mar 07 2021 *)

Formula

E.g.f.: exp(Sum_{j=1..5} x^j/j!) - exp(Sum_{j=1..4} x^j/j!).

A229248 Number of set partitions of {1,...,n} with largest set of size 6.

Original entry on oeis.org

1, 7, 56, 420, 3150, 24024, 187110, 1497210, 12321309, 104379275, 910501592, 8176340536, 75557540604, 718108992888, 7015008076980, 70388350377492, 724955013327237, 7658820319677219, 82939240748756392, 920067296840668900, 10448713239329294930
Offset: 6

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Links

Crossrefs

Column k=6 of A080510.

Programs

  • Maple
    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,6)-G(n,5):
seq(a(n), n=6..30);

Formula

E.g.f.: exp(Sum_{j=1..6} x^j/j!) - exp(Sum_{j=1..5} x^j/j!).

A229249 Number of set partitions of {1,...,n} with largest set of size 7.

Original entry on oeis.org

1, 8, 72, 600, 4950, 41184, 348348, 3008148, 26608725, 241395440, 2247683152, 21485909952, 210840271980, 2123351405280, 21937875152760, 232419281905272, 2523691371079725, 28070949453307992, 319668800125675000, 3725037254807468600, 44393091629344788330
Offset: 7

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Links

Crossrefs

Column k=7 of A080510.

Programs

  • Maple
    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,7)-G(n,6):
seq(a(n), n=7..30);

Formula

E.g.f.: exp(Sum_{j=1..7} x^j/j!) - exp(Sum_{j=1..6} x^j/j!).

A229250 Number of set partitions of {1,...,n} with largest set of size 8.

Original entry on oeis.org

1, 9, 90, 825, 7425, 66924, 609609, 5643495, 53275365, 513949865, 5072383602, 51247279161, 530162317815, 5616049288500, 60907673202255, 676114396423257, 7679484722988045, 89216205660482175, 1059689275333095550, 12863300130089805825, 159506272335882076605
Offset: 8

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Links

Crossrefs

Column k=8 of A080510.

Programs

  • Maple
    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,8)-G(n,7):
seq(a(n), n=8..30);

Formula

E.g.f.: exp(Sum_{j=1..8} x^j/j!) - exp(Sum_{j=1..7} x^j/j!).

A229251 Number of set partitions of {1,...,n} with largest set of size 9.

Original entry on oeis.org

1, 10, 110, 1100, 10725, 104104, 1016015, 10032880, 100643400, 1028142830, 10712984282, 113961363880, 1238298284860, 13747432565790, 155944562191220, 1807325391776872, 21396572780305250, 258695925902828700, 3193365395052825850, 40233167990427412000
Offset: 9

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Links

Crossrefs

Column k=9 of A080510.

Programs

  • Maple
    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)-G(n,8):
seq(a(n), n=9..30);

Formula

E.g.f.: exp(Sum_{j=1..9} x^j/j!) - exp(Sum_{j=1..8} x^j/j!).

A229252 Number of set partitions of {1,...,n} with largest set of size 10.

Original entry on oeis.org

1, 11, 132, 1430, 15015, 156156, 1625624, 17055896, 181158120, 1953517566, 21426984722, 239340203466, 2724654560628, 31626047962432, 374383760685660, 4520229252281160, 55662052670665870, 698975472340315170, 8949358054013356980, 116803043793523277190
Offset: 10

Views

Author

Alois P. Heinz, Sep 17 2013

Keywords

Links

Crossrefs

Column k=10 of A080510.

Programs

  • Maple
    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)-G(n,9):
seq(a(n), n=10..30);

Formula

E.g.f.: exp(Sum_{j=1..10} x^j/j!) - exp(Sum_{j=1..9} x^j/j!).

A339030 T(n, k) = Sum_{p in P(n, k)} card(p), where P(n, k) is the set of set partitions of {1,2,...,n} where the largest block has size k and card(p) is the number of blocks of p. Triangle T(n, k) for 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 8, 1, 0, 5, 85, 50, 10, 1, 0, 6, 300, 280, 75, 12, 1, 0, 7, 1071, 1540, 525, 105, 14, 1, 0, 8, 3976, 8456, 3570, 840, 140, 16, 1, 0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1
Offset: 0

Views

Author

Peter Luschny, Nov 22 2020

Keywords

Examples

			Triangle starts:
0: [1]
1: [0, 1]
2: [0, 2, 1]
3: [0, 3, 6,     1]
4: [0, 4, 24,    8,     1]
5: [0, 5, 85,    50,    10,    1]
6: [0, 6, 300,   280,   75,    12,   1]
7: [0, 7, 1071,  1540,  525,   105,  14,   1]
8: [0, 8, 3976,  8456,  3570,  840,  140,  16,  1]
9: [0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1]
.
T(4,0) = 0  = 0*card({})
T(4,1) = 4  = 4*card({1|2|3|4}).
T(4,2) = 24 = 3*card({12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34})
            + 2*card({12|34, 13|24, 14|23}).
T(4,3) = 8  = 2*card({123|4, 124|3, 134|2, 1|234}).
T(4,4) = 1  = 1*card({1234}).
.
Seen as the projection of a 2-dimensional statistic this is, for n = 6:
[  0   0    0     0     0    0   0]
[  0   0    0     0     0    0   6]
[  0   0    0    45   180   75   0]
[  0   0   20   180    80    0   0]
[  0   0   30    45     0    0   0]
[  0   0   12     0     0    0   0]
[  0   1    0     0     0    0   0]
The row sum projection gives row 6 of this triangle, and the column sum projection gives [0, 1, 62, 270, 260, 75, 6], which appears in a decapitated version as row 5 in A321331.

Crossrefs

Cf. A005493 with 1 prepended are the row sums.
Cf. A080510, A262071, A321331.

Programs

  • SageMath
    def A339030Row(n):
    if n == 0: return [1]
    M = matrix(n + 1)
    for k in (1..n):
        for p in SetPartitions(n):
            if p.max_block_size() == k:
                M[k, len(p)] += p.cardinality()
    return [sum(M[k, j] for j in (0..n)) for k in (0..n)]
for n in (0..9): print(A339030Row(n))
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