A283424 -id:A283424 - cp's OEIS frontend

A138378 Number of embedded coalitions in an n-person game.

1, 3, 10, 37, 151, 674, 3263, 17007, 94828, 562595, 3535027, 23430840, 163254885, 1192059223, 9097183602, 72384727657, 599211936355, 5150665398898, 45891416030315, 423145657921379, 4031845922290572, 39645290116637023, 401806863439720943, 4192631462935194064

Offset: 1

David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008

nonn

Same as A005493, apart from offset. - R. J. Mathar, Sep 23 2011

The strategic behavior of players depends crucially on the coalition structures of a game.

			a(1) = combination(1,0) = 1,
a(2) = combination(2,1) + combination(2,0)= 3,
a(3) = combination(3,2)* a(1) + combination(3,2) + combination(3,1) + combination(3,0)= 10,
a(4) = combination(4,3)* {a(1) + a(2)} + combination(4,2)* a(1) + combination7(4,3)combination(4,2) + combination(4,1) + combination(4,0)= 37,
a(5) = combination(5,4)* {a(1) + a(2) + a(3)} combination(5,3)* {a(1) + a(2)} + combination(5,2)* a(1) + combination(5,4) + combination(5,3) + combination(5,2) + combination(5,1) + combination(5,0)= 151.

J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, New York, 1995.

Alois P. Heinz, Table of n, a(n) for n = 1..575
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Gábor Czédli, Lattices with lots of congruence energy, arXiv:2205.02294 [math.RA], 2022.
A. L. L. Gao, S. Kitaev, and P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
David W. K. Yeung, Recursive sequence identifying the number of embedded coalitions, International Game Theory Review 10(2008), 129-136.

Cf. A000110, A005493, A048993, A138379.

Column k=1 of A283424.

[&+[k*StirlingSecond(n, k): k in [1..n]]: n in [1..25]]; // Vincenzo Librandi, May 18 2019

b:= proc(n, m) option remember;
     `if`(n=0, m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Dec 10 2024

a[n_] := Sum[Binomial[n, j] BellB[j], {j, 0, n-1}];
Array[a, 24] (* Jean-François Alcover, Aug 19 2018 *)

a(1) = combination(1,0) = 1, a(2) = combination(2,1) + combination(2,0)= 3, a(n) = {SUM(i=2 to n-1) combination(n,i)} * {SUM(j=1 to i-1) a(n)} + SUM(i=0 to n-1) combination(n,i), for n > 2.
G.f.: 1/U(0) where U(k)= 1 - x*(k+3) - x^2*(k+1)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: 1/(U(0)-x) where U(k)= 1 - x - x*(k+1)/(1 - x/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2012
G.f.: -G(0)/x^2 where G(k) = 1 - 1/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 08 2013
G.f.: Q(0)/x^2 -1/x^2, where Q(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*(k+1))*(1-x*(k+2))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
a(n) = Bell(n+1)-Bell(n) = Sum_{k=1..n} k*Stirling2(n,k). - Alois P. Heinz, May 11 2017
E.g.f.: (exp(x) - 1) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jan 26 2020

A285595 Sum T(n,k) of the k-th entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 4, 2, 17, 10, 3, 76, 52, 18, 4, 362, 274, 111, 28, 5, 1842, 1500, 675, 200, 40, 6, 9991, 8614, 4185, 1380, 325, 54, 7, 57568, 51992, 26832, 9568, 2510, 492, 70, 8, 351125, 329650, 178755, 67820, 19255, 4206, 707, 88, 9, 2259302, 2192434, 1239351, 494828, 149605, 35382, 6629, 976, 108, 10

Offset: 1

Alois P. Heinz, Apr 22 2017

T(n,k) is also k times the number of blocks of size >k in all set partitions of [n+1]. T(3,2) = 10 = 2 * 5 because there are 5 blocks of size >2 in all set partitions of [4], namely in 1234, 123|4, 124|3, 134|2, 1|234.

			T(3,2) = 10 because the sum of the second entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 2+2+3+3+0  = 10.
Triangle T(n,k) begins:
      1;
      4,     2;
     17,    10,     3;
     76,    52,    18,    4;
    362,   274,   111,   28,    5;
   1842,  1500,   675,  200,   40,   6;
   9991,  8614,  4185, 1380,  325,  54,  7;
  57568, 51992, 26832, 9568, 2510, 492, 70, 8;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Partition of a set

Column k=1 gives A124325(n+1).
Row sums give A000110(n) * A000217(n) = A105488(n+3).
Main diagonal and first lower diagonal give: A000027, A028552.
Cf. A007318, A175757, A278677, A283424, A285362, A285793, A286897.

T:= proc(h) option remember; local b; b:=
      proc(n, l) option remember; `if`(n=0, [1, 0],
        (p-> p+[0, (h-n+1)*p[1]*x^1])(b(n-1, [l[], 1]))+
         add((p-> p+[0, (h-n+1)*p[1]*x^(l[j]+1)])(b(n-1,
         sort(subsop(j=l[j]+1, l), `>`))), j=1..nops(l)))
      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2])
    end:
seq(T(n), n=1..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0],
      add((p-> p+[0, p[1]*add(x^k, k=1..j-1)])(
         b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i)*i, i=1..n))(b(n+1)[2]):
seq(T(n), n=1..12);

b[n_] := b[n] = If[n == 0, {1, 0}, Sum[# + {0, #[[1]]*Sum[x^k, {k, 1, j-1} ]}&[b[n - j]*Binomial[n - 1, j - 1]], {j, 1, n}]];
T[n_] := Table[Coefficient[#, x, i]*i, {i, 1, n}] &[b[n + 1][[2]]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, May 23 2018, translated from 2nd Maple program *)

T(n,k) = k * Sum_{j=k+1..n+1} binomial(n+1,j)*A000110(n+1-j).
T(n,k) = k * Sum_{j=k+1..n+1} A175757(n+1,j).
Sum_{k=1..n} T(n,k)/k = A278677(n-1).

A124325 Number of blocks of size >1 in all partitions of an n-set.

Original entry on oeis.org

0, 0, 1, 4, 17, 76, 362, 1842, 9991, 57568, 351125, 2259302, 15288000, 108478124, 805037105, 6233693772, 50257390937, 421049519856, 3659097742426, 32931956713294, 306490813820239, 2945638599347760, 29198154161188501

Offset: 0

Emeric Deutsch, Oct 28 2006

nonn

Sum of the first entries in all blocks of all set partitions of [n-1]. a(4) = 17 because the sum of the first entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+4+3+3+6 = 17. - Alois P. Heinz, Apr 24 2017

			a(3) = 4 because in the partitions 123, 12|3, 13|2, 1|23, 1|2|3 we have four blocks of size >1.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000110, A123324, A124327, A285595.

Column k=2 of A283424.

with(combinat): c:=n->bell(n+1)-bell(n)-n*bell(n-1): seq(c(n),n=0..23);

nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x]-1-x)Exp[Exp[x]-1],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 28 2013 *)

N = 66;  x = 'x + O('x^N);
egf = (exp(x)-1-x)*exp(exp(x)-1) + 'c0;
gf = serlaplace(egf);
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Mar 29 2013 */

a(n) = B(n+1)-B(n)-n*B(n-1), where B(q) are the Bell numbers (A000110).
E.g.f.: (exp(z)-1-z)*exp(exp(z)-1).
a(n) = Sum_{k=0..floor(n/2)} k*A124324(n,k).
a(n) = A285595(n-1,1). - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=1..n*(n-1)/2} k * A124327(n-1,k) for n>1. - Alois P. Heinz, Dec 05 2023

A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

Original entry on oeis.org

1, 1, 3, 8, 28, 107, 459, 2151, 10931, 59700, 348146, 2155925, 14112377, 97266301, 703484851, 5323515156, 42040470092, 345670438963, 2953171501547, 26166317121747, 240047041176843, 2276607815242880, 22290187889601330, 225018607554567149, 2339331996135377345

Offset: 1

Geoffrey Critzer, Apr 02 2013

nonn

			a(4) = 8 because we have: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4},{3}}, {{1},{2},{3},{4}}.

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A000110, A003724, A005046, A102286, A124322, A131719, A283424.

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`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
    end:
a:= n-> (p-> add(coeff(p, x, i)*(i+1), i=0..degree(p)))(b(n-1$2)):
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 08 2015
# second Maple program:
b:= proc(n, t, m) option remember; `if`(n=0, t, (m-1)*
      b(n-1, t, m)+b(n-1, 1-t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n-1, 1$2):
seq(a(n), n=1..25);  # Alois P. Heinz, May 17 2023

nn=25;Drop[Range[0,nn]!CoefficientList[Series[Integrate[Exp[Cosh[x]-1]D[ Exp[Sinh[x]],x],x],{x,0,nn}],x],1]

E.g.f. A(x) satisfies: A'(x) = B'(x)*C(x) where B(x) is the e.g.f. for A003724 and C(x) is the e.g.f. for A005046.
a(n) = Sum_{k=0..floor((n-1)/2)} (k+1)*A124322(n-1,k). - Alois P. Heinz, Apr 02 2013
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * Bell(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022
From Alois P. Heinz, May 17 2023: (Start)
a(n) = Sum_{k=0..n-1} (-1)^k * A283424(n-1,k).
a(n) mod 2 = A131719(n+1). (End)

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

A286896 Number of blocks of size >= n in all set partitions of [2n].

Original entry on oeis.org

1, 3, 17, 137, 1395, 16955, 237426, 3740609, 65197797, 1241499241, 25577181324, 565688751435, 13346516581331, 334144326030052, 8837737924901855, 245998212661731213, 7182425756528424275, 219332432679783740235, 6987451758608249737342, 231704015156531645221237

Offset: 0

Alois P. Heinz, May 15 2017

nonn

			a(2) = 17: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34. Here three set partitions contain 2 blocks of size 2.

Alois P. Heinz, Table of n, a(n) for n = 0..445
Wikipedia, Partition of a set

Cf. A000110, A283424.

b:= proc(n, k) option remember; `if`(k>n, 0,
      binomial(n, k)*combinat[bell](n-k)+b(n, k+1))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..25);

a[n_] := Sum[Binomial[2 n, j] BellB[j], {j, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 28 2018 *)

a(n) = Sum_{j=0..n} binomial(2n,j) * Bell(j).
a(n) = A283424(2n,n).
a(n) ~ 2^(2*n) * exp(n/LambertW(n) - n - 1) * n^(n - 1/2) / (sqrt(Pi*(1 + LambertW(n))) * LambertW(n)^n). - Vaclav Kotesovec, Jul 23 2021

A288786 Number of blocks of size >= four in all set partitions of n.

Original entry on oeis.org

1, 6, 37, 225, 1395, 8944, 59585, 413117, 2981310, 22380814, 174600298, 1413841252, 11868587577, 103155618776, 927141821215, 8606806236367, 82430269073469, 813600584094320, 8267450613029789, 86406853732930699, 927993270700444588, 10232636504064477996

Offset: 4

Alois P. Heinz, Jun 15 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..575
Wikipedia, Partition of a set

Column k=4 of A283424.

Cf. A000110.

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, 4):
seq(a(n), n=4..30);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+[0,
     `if`(j>3, p[1], 0)])(b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=4..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, 4];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

a(n) = Bell(n+1) - Sum_{j=0..3} binomial(n,j) * Bell(n-j).
a(n) = Sum_{j=0..n-4} binomial(n,j) * Bell(j).
E.g.f.: (exp(x) - Sum_{k=0..3} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 25 2022

A288787 Number of blocks of size >= five in all set partitions of n.

Original entry on oeis.org

1, 7, 50, 345, 2392, 16955, 123707, 932010, 7260709, 58509323, 487593202, 4199841037, 37361858716, 342989895895, 3246458915947, 31653980371254, 317654338317380, 3278058775976704, 34757921507150964, 378372365291381716, 4225533329681577846, 48375204740642752562

Offset: 5

Alois P. Heinz, Jun 15 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..575
Wikipedia, Partition of a set

Column k=5 of A283424.

Cf. A000110.

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, 5):
seq(a(n), n=5..30);

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, 5];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

a(n) = Bell(n+1) - Sum_{j=0..4} binomial(n,j) * Bell(n-j).
a(n) = Sum_{j=0..n-5} binomial(n,j) * Bell(j).
E.g.f.: (exp(x) - Sum_{k=0..4} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 26 2022

A288788 Number of blocks of size >= 6 in all set partitions of n.

Original entry on oeis.org

1, 8, 65, 502, 3851, 29921, 237426, 1932529, 16173029, 139320277, 1235847277, 11288120480, 106132359679, 1026681599731, 10212591089574, 104393925768077, 1095895294558168, 11806719056706773, 130457490607638988, 1477428802636263486, 17138268233851671782

Offset: 6

Alois P. Heinz, Jun 15 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..575
Wikipedia, Partition of a set

Column k=6 of A283424.

Cf. A000110.

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, 6):
seq(a(n), n=6..30);

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, 6];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

a(n) = Bell(n+1) - Sum_{j=0..5} binomial(n,j) * Bell(n-j).
a(n) = Sum_{j=0..n-6} binomial(n,j) * Bell(j).
E.g.f.: (exp(x) - Sum_{k=0..5} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 26 2022

A288789 Number of blocks of size >= 7 in all set partitions of n.

Original entry on oeis.org

1, 9, 82, 701, 5897, 49854, 427597, 3740609, 33479542, 307119477, 2890138160, 27911144971, 276632735047, 2813333368854, 29349063282197, 313940448544057, 3441759044602385, 38652680805862224, 444450158120668786, 5229815283321976222, 62942722623990478840

Offset: 7

Alois P. Heinz, Jun 15 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..575
Wikipedia, Partition of a set

Column k=7 of A283424.

Cf. A000110.

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, 7):
seq(a(n), n=7..30);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+
      `if`(j>6, [0, p[1]], 0))(b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=7..30);  # Alois P. Heinz, Jun 26 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, 7];
Table[a[n], {n, 7, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)

a(n) = Bell(n+1) - Sum_{j=0..6} binomial(n,j) * Bell(n-j).
a(n) = Sum_{j=0..n-7} binomial(n,j) * Bell(j).
E.g.f.: (exp(x) - Sum_{k=0..6} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 26 2022

cp's OEIS Frontend

A138378 Number of embedded coalitions in an n-person game.

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A285595 Sum T(n,k) of the k-th entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A124325 Number of blocks of size >1 in all partitions of an n-set.

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A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

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

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A286896 Number of blocks of size >= n in all set partitions of [2n].

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A288786 Number of blocks of size >= four in all set partitions of n.

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