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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 Alois P. Heinz, Table of n, a(n) for n = 0..200 F. Cellarosi and Ya. G. Sinai, The Möbius function and statistical mechanics, Bull. Math. Sci., 2011. Wikipedia, Exponential integral 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. Cf. A000110, A000248, A001620, A322364, A322365, A322380, A322381, A323339, A323340. Sequence in context: A082387 A032923 A018750 * A370106 A147845 A077778 Adjacent sequences: A177205 A177206 A177207 * A177209 A177210 A177211 KEYWORD frac,nonn AUTHOR Franklin T. Adams-Watters, May 04 2010 STATUS approved

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