This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A177208 #33 Nov 09 2022 12:04:30
%S A177208 1,1,3,17,19,81,8351,184553,52907,1768847,70442753,1096172081,
%T A177208 22198464713,195894185831,42653714271997,30188596935106763,
%U A177208 20689743895700791,670597992748852241,71867806446352961329,8445943795439038164379,379371134635840861537
%N A177208 Numerators of exponential transform of 1/n.
%C A177208 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 A177208 Numerators of moments of Dickman-De Bruijn distribution as shown on page 257 of Cellarosi and Sinai. [_Jonathan Vos Post_, Jan 07 2012]
%D A177208 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 A177208 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 A177208 Alois P. Heinz, <a href="/A177208/b177208.txt">Table of n, a(n) for n = 0..200</a>
%H A177208 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 A177208 Wikipedia, <a href="http://en.wikipedia.org/wiki/Exponential_integral">Exponential integral</a>
%F A177208 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 A177208 Knuth & Trabb-Pardo (6.7) gives a recurrence. - _N. J. A. Sloane_, Nov 09 2022
%e A177208 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 A177208 b:= proc(n) option remember; `if`(n=0, 1,
%p A177208 add(binomial(n-1, j-1)*b(n-j)/j, j=1..n))
%p A177208 end:
%p A177208 a:= n-> numer(b(n)):
%p A177208 seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 08 2012
%t A177208 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 A177208 (PARI) Vec(serlaplace(exp(sum(n=1,30,x^n/(n*n!),O(x^31)))))
%Y A177208 Denominators are in A177209.
%Y A177208 Cf. A000110, A000248, A001620, A322364, A322365, A322380, A322381, A323339, A323340.
%K A177208 frac,nonn
%O A177208 0,3
%A A177208 _Franklin T. Adams-Watters_, May 04 2010