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A288789
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Number of blocks of size >= 7 in all set partitions of n.
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2
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1, 9, 82, 701, 5897, 49854, 427597, 3740609, 33479542, 307119477, 2890138160, 27911144971, 276632735047, 2813333368854, 29349063282197, 313940448544057, 3441759044602385, 38652680805862224, 444450158120668786, 5229815283321976222, 62942722623990478840
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OFFSET
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7,2
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LINKS
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FORMULA
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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
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MAPLE
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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]:
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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