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A288789 Number of blocks of size >= 7 in all set partitions of n. 2
1, 9, 82, 701, 5897, 49854, 427597, 3740609, 33479542, 307119477, 2890138160, 27911144971, 276632735047, 2813333368854, 29349063282197, 313940448544057, 3441759044602385, 38652680805862224, 444450158120668786, 5229815283321976222, 62942722623990478840 (list; graph; refs; listen; history; text; internal format)
OFFSET

7,2

LINKS

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

Wikipedia, Partition of a set

FORMULA

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

MAPLE

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<k, 0,

      g(n, k+1) +binomial(n, k)*b(n-k))

    end:

a:= 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

MATHEMATICA

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-Fran├žois Alcover, May 28 2018, from Maple *)

CROSSREFS

Column k=7 of A283424.

Cf. A000110.

Sequence in context: A275917 A293803 A263817 * A033119 A033127 A099371

Adjacent sequences:  A288786 A288787 A288788 * A288790 A288791 A288792

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 15 2017

STATUS

approved

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Last modified August 8 09:40 EDT 2022. Contains 356009 sequences. (Running on oeis4.)