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A177208
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Numerators of exponential transform of 1/n.
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8
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1, 1, 3, 17, 19, 81, 8351, 184553, 52907, 1768847, 70442753, 1096172081, 22198464713, 195894185831, 42653714271997, 30188596935106763, 20689743895700791, 670597992748852241, 71867806446352961329, 8445943795439038164379, 379371134635840861537
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OFFSET
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0,3
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COMMENTS
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b(n) = a(n)/A177209(n) is the sum over all set partitions of [n] of the product of the reciprocals of the part sizes.
Numerators of moments of Dickman-De Bruijn distribution as shown on page 257 of Cellarosi and Sinai. [Jonathan Vos Post, Jan 07 2012]
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), pp. 228-230.
Knuth, Donald E., and Luis Trabb Pardo. "Analysis of a simple factorization algorithm." Theoretical Computer Science 3.3 (1976): 321-348. See Eq. (6.6) and (6.7), page 334.
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LINKS
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FORMULA
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E.g.f. for fractions is exp(f(z)), where f(z) = sum(k>0, z^k/(k*k!)) = integral(0..z,(exp(t)-1)/t dt) = Ei(z) - gamma - log(z) = -Ein(-z). Here gamma is Euler's constant, and Ei and Ein are variants of the exponential integral.
Knuth & Trabb-Pardo (6.7) gives a recurrence. - N. J. A. Sloane, Nov 09 2022
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EXAMPLE
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For n=4, there is 1 set partition with a single part of size 4, 4 with sizes [3,1], 3 with sizes [2,2], 6 with sizes [2,1,1], and 1 with sizes [1,1,1,1]; so b(4) = 1/4 + 4/(3*1) + 3/(2*2) + 6/(2*1*1) + 1/(1^4) = 1/4 + 4/3 + 3/4 + 3 + 1 = 19/4.
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1,
add(binomial(n-1, j-1)*b(n-j)/j, j=1..n))
end:
a:= n-> numer(b(n)):
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MATHEMATICA
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b[n_] := b[n] = If[n==0, 1, Sum[Binomial[n-1, j-1]*b[n-j]/j, {j, 1, n}]]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)
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PROG
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(PARI) Vec(serlaplace(exp(sum(n=1, 30, x^n/(n*n!), O(x^31)))))
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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