A355144 -id:A355144 - cp's OEIS frontend

A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)).

1, 1, 2, 4, 1, 11, 4, 32, 20, 113, 80, 10, 422, 385, 70, 1788, 1792, 560, 8015, 9492, 3360, 280, 39435, 50640, 23100, 2800, 204910, 295020, 147840, 30800, 1144377, 1763300, 1044120, 246400, 15400, 6722107, 11278410, 7241520, 2202200, 200200, 41877722

Offset: 0

Emeric Deutsch, Nov 14 2006

Row n contains 1+floor(n/3) terms. Row sums yield the Bell numbers (A000110). T(n,0)=A124504(n). Sum(k*T(n,k), k=0..floor(n/3))=A105480(n+1).

			T(4,1)=4 because we have 1|234, 134|2, 124|3 and 123|4.
Triangle starts:
    1;
    1;
    2;
    4,   1;
   11,   4;
   32,  20;
  113,  80, 10;
  422, 385, 70;
  ...

Alois P. Heinz, Rows n = 0..250, flattened

Cf. A000110, A124504, A105480, A355144.

T(3n,n) gives A025035.

G:=exp(exp(z)-1+(t-1)*z^3/6): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/3)) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(i=3, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015

nn = 8; k = 3; Range[0, nn]! CoefficientList[Series[Exp[Exp[x] - 1 + (y - 1) x^k/k!], {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 26 2012 *)

E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)z^3/6).

A288785 Number of blocks of size >= three in all set partitions of n.

Original entry on oeis.org

1, 5, 26, 137, 750, 4307, 25996, 164825, 1096217, 7633650, 55549664, 421599778, 3331027887, 27349472297, 232967157736, 2055635993935, 18762063976810, 176896220650029, 1720762736285790, 17249873608817569, 178010337967774511, 1889129778601708612

Offset: 3

Alois P. Heinz, Jun 15 2017

nonn

			a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
a(5) = 26: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 125|3|4, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 135|2|4, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
a(6) = 137: 123456, 12345|6, 12346|5, ..., 123|456, 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234, ..., 1|256|3|4, 1|2|356|4, 1|2|3|456.

Alois P. Heinz, Table of n, a(n) for n = 3..575
Wikipedia, Partition of a set

Column k=3 of A283424.

Cf. A000110, A355144.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
g:= proc(n, k) option remember; `if`(n g(n, 3):
seq(a(n), n=3..30);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+[0,
     `if`(j>2, p[1], 0)])(b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=3..30);  # Alois P. Heinz, Jan 06 2022

b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1], {j, 1, n}]];
g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k+1] + Binomial[n, k]*b[n - k]];
a[n_] := g[n, 3];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

a(n) = Bell(n+1) - Sum_{j=0..2} binomial(n,j) * Bell(n-j).
a(n) = Sum_{j=0..n-3} binomial(n,j) * Bell(j).
a(n) = Sum_{k=1..n} k * A355144(n,k). - Alois P. Heinz, Jun 20 2022
E.g.f.: (exp(x) - 1 - x - x^2/2) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 24 2022

cp's OEIS Frontend

A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula