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A288792
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Number of blocks of size >= ten in all set partitions of n.
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2
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1, 12, 145, 1600, 17032, 179132, 1883117, 19929390, 213332101, 2316793121, 25577181324, 287421068697, 3290394397097, 38393883291996, 456753452800691, 5540597439008861, 68530489547341697, 864218608315007230, 11109867095322262250, 145563654356205885737
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OFFSET
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10,2
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LINKS
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FORMULA
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a(n) = Bell(n+1) - Sum_{j=0..9} binomial(n,j) * Bell(n-j).
a(n) = Sum_{j=0..n-10} binomial(n,j) * Bell(j).
E.g.f.: (exp(x) - Sum_{k=0..9} 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, 10):
seq(a(n), n=10..30);
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MATHEMATICA
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Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 10}], {n, 10, 30}] (* Indranil Ghosh, Jul 06 2017 *)
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PROG
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(Python)
from sympy import bell, binomial
def a(n): return sum([binomial(n, j)*bell(j) for j in range(n - 9)])
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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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