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 A005727 M0868 #59 Oct 27 2023 19:44:01
%S A005727 1,1,2,3,8,10,54,-42,944,-5112,47160,-419760,4297512,-47607144,
%T A005727 575023344,-7500202920,105180931200,-1578296510400,25238664189504,
%U A005727 -428528786243904,7700297625889920,-146004847062359040,2913398154375730560,-61031188196889482880
%N A005727 n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers.
%D A005727 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, table at foot of page.
%D A005727 G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.
%D A005727 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005727 Alois P. Heinz, <a href="/A005727/b005727.txt">Table of n, a(n) for n = 0..400</a> (first 101 terms from T. D. Noe)
%H A005727 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 36.5, "The function x^x"
%H A005727 H. W. Gould, <a href="http://dx.doi.org/10.1216/rmjm/1181072076">A Set of Polynomials Associated with the Higher Derivatives of y=xxy=x^x</a>, Rocky Mountain J. Math. 26(2) 1996.
%H A005727 R. K. Guy, <a href="/A005727/a005727.pdf">Letter to N. J. A. Sloane, 1986</a>
%H A005727 R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H A005727 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H A005727 G. H. Hardy, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM1516.0001.001">A Course of Pure Mathematics</a>, Cambridge, The University Press, 1908.
%H A005727 D. H. Lehmer, <a href="http://dx.doi.org/10.1216/RMJ-1985-15-2-461">Numbers associated with Stirling Numbers and x^x</a>, Rocky Mountain J. Math., 15(2) 1985, p. 461.
%H A005727 R. R. Patterson and G. Suri, <a href="/A005168/a005168.pdf">The derivatives of x^x</a>, date unknown. Preprint. [Annotated scanned copy]
%F A005727 For n>0, a(n) = Sum_{k=0..n} b(n, k), where b(n, k) is a Lehmer-Comtet number of the first kind (see A008296).
%F A005727 E.g.f.: (1+x)^(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*A000248(k). - _Vladeta Jovovic_, Oct 02 2003
%F A005727 From _Mélika Tebni_, May 22 2022: (Start)
%F A005727 a(0) = 1, a(n) = a(n-1)+Sum_{k=0..n-2} (-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*a(k).
%F A005727 a(n) = Sum_{k=0..n} (-1)^(n-k)*A293297(k)*binomial(n, k).
%F A005727 a(n) = Sum_{k=0..n} (-1)^k*A203852(k)*binomial(n, k). (End)
%p A005727 A005727 := proc(n) option remember; `if`(n=0, 1, A005727(n-1)+add((-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*A005727(k), k=0..n-2)) end:
%p A005727 seq(A005727(n), n=0..23); # _Mélika Tebni_, May 22 2022
%t A005727 NestList[ Factor[ D[ #1, x ] ]&, x^x, n ] /. (x->1)
%t A005727 Range[0, 22]! CoefficientList[ Series[(1 + x)^(1 + x), {x, 0, 22}], x] (* _Robert G. Wilson v_, Feb 03 2013 *)
%o A005727 (PARI) a(n)=if(n<0,0,n!*polcoeff((1+x+x*O(x^n))^(1+x),n))
%Y A005727 Row sums of A008296. Column k=2 of A215703 and of A277537.
%Y A005727 Cf. A005168, A176118, A293297, A203852.
%K A005727 sign,easy,nice
%O A005727 0,3
%A A005727 _N. J. A. Sloane_