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A177208 Numerators of exponential transform of 1/n. 8
1, 1, 3, 17, 19, 81, 8351, 184553, 52907, 1768847, 70442753, 1096172081, 22198464713, 195894185831, 42653714271997, 30188596935106763, 20689743895700791, 670597992748852241, 71867806446352961329, 8445943795439038164379, 379371134635840861537 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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]
REFERENCES
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.
LINKS
F. Cellarosi and Ya. G. Sinai, The Möbius function and statistical mechanics, Bull. Math. Sci., 2011.
FORMULA
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
EXAMPLE
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.
MAPLE
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)):
seq(a(n), n=0..25); # Alois P. Heinz, Jan 08 2012
MATHEMATICA
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 *)
PROG
(PARI) Vec(serlaplace(exp(sum(n=1, 30, x^n/(n*n!), O(x^31)))))
CROSSREFS
Denominators are in A177209.
Sequence in context: A082387 A032923 A018750 * A370106 A147845 A077778
KEYWORD
frac,nonn
AUTHOR
STATUS
approved

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Last modified May 25 11:02 EDT 2024. Contains 372788 sequences. (Running on oeis4.)