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A361380
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Sum over the j-th term of the (n-j)-th inverse binomial transform of the Bell numbers (A000110) for all j in [n].
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3
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1, 2, 3, 6, 17, 56, 215, 922, 4305, 21894, 119539, 696632, 4314925, 28237146, 194602079, 1407456694, 10649642837, 84100177424, 691474151187, 5907288773554, 52340230286509, 480153099982726, 4553711640946919, 44584683333637168, 450075389309517849
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graph;
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listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{i=0..n} i! * [x^i] exp(exp(x)-(n-i)*x-1).
a(n) = Sum_{0<=j<=i<=n} binomial(i,j)*(i-n)^(i-j)*Bell(j).
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MAPLE
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a:= n-> add(add(binomial(i, j)*(i-n)^(i-j)*combinat[bell](j), j=0..i), i=0..n):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> add(i!*coeff(series(exp(exp(x)-(n-i)*x-1), x, i+1), x, i), i=0..n):
seq(a(n), n=0..25);
# third Maple program:
b:= proc(n, m) option remember;
`if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
end:
a:= n-> add(b(i, i-n), i=0..n):
seq(a(n), n=0..25);
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PROG
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(Python)
from math import comb
from sympy import bell
def A361380(n): return sum(comb(i, j)*(i-n)**(i-j)*bell(j) for i in range(n+1) for j in range(i+1)) # Chai Wah Wu, Apr 05 2023
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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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