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A134980
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a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000110(k).
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10
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1, 2, 10, 77, 799, 10427, 163967, 3017562, 63625324, 1512354975, 40012800675, 1166271373797, 37134022033885, 1282405154139046, 47745103281852282, 1906411492286148245, 81267367663098939459, 3683790958912910588623, 176937226305157687076779
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OFFSET
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0,2
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COMMENTS
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Number of partitions of [2n] where at least n blocks contain their own index element. a(2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, Jan 07 2022
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LINKS
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FORMULA
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a(n) = exp(-1)*Sum_{k>=0} (n+k)^n/k!.
E.g.f.: A(x) = exp(-1)*Sum_{k>=0} (1+k*x)^k/k!.
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MAPLE
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with(combinat): a:= n-> add(binomial(n, k)*n^(n-k)*bell(k), k=0..n):
# Alternate:
g:= proc(n) local S;
S:= series(exp(exp(x)+n*x-1), x, n+1);
n!*coeff(S, x, n);
end proc:
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
k*b(n-1, k)+ b(n-1, k+1))
end:
a:= n-> b(n$2):
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MATHEMATICA
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Join[{1}, Table[Sum[Binomial[n, k]*n^(n-k)*BellB[k], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jun 09 2020 *)
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PROG
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(Sage)
return add(binomial(n, k)*n^(n-k)*bell_number(k) for k in (0..n))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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