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A188634
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E.g.f.: Sum_{n>=0} (1 - exp(-(n+1)*x))^n/(n+1).
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5
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1, 1, 4, 46, 1066, 41506, 2441314, 202229266, 22447207906, 3216941445106, 578333776748674, 127464417117501586, 33800841048945424546, 10617398393395844992306, 3898852051843774954576834, 1654948033478889053351543506, 804119629083230641164978005986
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{j=0..n} (j+1)^(n-1) * Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n.
Ignoring the initial term, equals a diagonal of array A099594, which forms the poly-Bernoulli numbers B(-k,n).
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 4*x^2/2! + 46*x^3/3! + 1066*x^4/4! + 41506*x^5/5! +...
where
A(x) = 1 + (1-exp(-2*x))/2 + (1-exp(-3*x))^2/3 + (1-exp(-4*x))^3/4 + (1-exp(-5*x))^4/5 + (1-exp(-6*x))^5/6 +...
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MATHEMATICA
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Table[Sum[(-1)^(k+n)*(k+1)^(n-1)*k!*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
Table[n!*SeriesCoefficient[Sum[(1-E^(-x*(k+1)))^k/(k+1), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 30 2012 *)
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PROG
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(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-(k+1)*x+x*O(x^n)))^k/(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=sum(j=0, n, (j+1)^(n-1)*sum(i=0, j, (-1)^(n+j-i)*binomial(j, i)*(j-i)^n))}
for(n=0, 20, print1(a(n), ", "))
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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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