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%I #52 Nov 02 2023 04:15:42
%S 1,-2,2,-4,-4,-48,-336,-2928,-28144,-298528,-3454432,-43286528,
%T -583835648,-8433987584,-129941213184,-2127349165824,-36889047574272,
%U -675548628690432,-13030733384956416,-264111424634864640
%N Revert transform of factorials n! (n >= 1).
%C First diagonal of triangle in A059370.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.
%H Vaclav Kotesovec, <a href="/A059372/b059372.txt">Table of n, a(n) for n = 1..400</a> (first 100 terms from T. D. Noe)
%H M. H. Albert, M. D. Atkinson and M. Klazar, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Albert/albert.html">The Enumeration of Simple Permutations</a>, J. Integer Seqs., Vol. 6, 2003.
%H Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, <a href="https://arxiv.org/abs/2303.13115">Blockwise simple permutations</a>, arXiv:2303.13115 [math.CO], 2023.
%H Emeric Deutsch and B. E. Sagan, <a href="https://arxiv.org/abs/math/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F a(n) ~ -exp(-2) * n! * (1 - 4/n + 2/n^2 - 34/(3*n^3) - 296/(3*n^4) - 4818/(5*n^5) - 508532/(45*n^6)). - _Vaclav Kotesovec_, Aug 04 2015
%F G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k! * A(x)^k. - _Ilya Gutkovskiy_, Apr 22 2020
%p # From Transforms, see the footer of the page.
%p REVERT([seq(k!, k=1..20)]); # _Peter Luschny_, May 01 2021
%p # Using function CompInv from A357588.
%p CompInv(10, n -> factorial(n)); # _Peter Luschny_, Oct 09 2022
%t nmax = 20; t[n_, k_] := t[n, k] = Sum[(m + 1)!*t[n - m - 1, k - 1], {m, 0, n - k}]; t[n_, 1] = n!; t[n_, n_] = 1; tnk = Table[t[n, k], {n, 1, nmax}, {k, 1, nmax}]; Inverse[tnk][[All, 1]] (* _Jean-François Alcover_, Jul 13 2016 *)
%Y Cf. A000142, A059370.
%K sign,easy
%O 1,2
%A _N. J. A. Sloane_, Jan 28 2001
%E More terms from _Vladeta Jovovic_, Mar 05 2001
%E Definition refined by _Georg Fischer_, May 01 2021
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