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Numerators of exponential transform of 1/n.
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%I #33 Nov 09 2022 12:04:30

%S 1,1,3,17,19,81,8351,184553,52907,1768847,70442753,1096172081,

%T 22198464713,195894185831,42653714271997,30188596935106763,

%U 20689743895700791,670597992748852241,71867806446352961329,8445943795439038164379,379371134635840861537

%N Numerators of exponential transform of 1/n.

%C 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.

%C Numerators of moments of Dickman-De Bruijn distribution as shown on page 257 of Cellarosi and Sinai. [_Jonathan Vos Post_, Jan 07 2012]

%D 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.

%D 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.

%H Alois P. Heinz, <a href="/A177208/b177208.txt">Table of n, a(n) for n = 0..200</a>

%H F. Cellarosi and Ya. G. Sinai, <a href="http://dx.doi.org/10.1007/s13373-011-0011-6">The Möbius function and statistical mechanics</a>, Bull. Math. Sci., 2011.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Exponential_integral">Exponential integral</a>

%F 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.

%F Knuth & Trabb-Pardo (6.7) gives a recurrence. - _N. J. A. Sloane_, Nov 09 2022

%e 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.

%p b:= proc(n) option remember; `if`(n=0, 1,

%p add(binomial(n-1, j-1)*b(n-j)/j, j=1..n))

%p end:

%p a:= n-> numer(b(n)):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 08 2012

%t 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_ *)

%o (PARI) Vec(serlaplace(exp(sum(n=1,30,x^n/(n*n!),O(x^31)))))

%Y Denominators are in A177209.

%Y Cf. A000110, A000248, A001620, A322364, A322365, A322380, A322381, A323339, A323340.

%K frac,nonn

%O 0,3

%A _Franklin T. Adams-Watters_, May 04 2010