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A327006 a(n) = A327005(n, n).

%I A327006 #14 Nov 17 2022 07:20:43
%S A327006 1,0,1,2,6,24,105,510,2765,16408,105210,724580,5330149,41649828,
%T A327006 344120777,2995027126,27368953170,261825429024,2615385871053,
%U A327006 27216432127818,294443944669341,3305528914953420,38442535155671262,462431164589185924,5745587267806107545
%N A327006 a(n) = A327005(n, n).
%H A327006 G. C. Greubel, <a href="/A327006/b327006.txt">Table of n, a(n) for n = 1..540</a>
%H A327006 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%H A327006 Retoz, <a href="http://retoz.free.fr/mai2021.pdf">Sous-dérangements</a>, Mai 2021.
%F A327006 a(n) = Sum_{k=1..n} BM[n][k] where BM is the BellMatrix(j -> j mod n) as defined in A264428.
%F A327006 Assuming offset = 0:
%F A327006 a(n) = n! * [x^n] exp(1 + (x - 1)*exp(x)). - Contributed by C. L. Martin (Retoz), May 16 2021.
%F A327006 a(n) = Sum_{j=0..n} binomial(n, j) * (-2)^(n-j) * A005387(j). - _G. C. Greubel_, Nov 17 2022
%p A327006 # BellMatrix is defined in A264428.
%p A327006 a := proc(n) BellMatrix(j -> modp(j, n), n): add(i, i in %[n]) end:
%p A327006 seq(a(n), n=1..25);
%p A327006 # Or, assuming offset = 0:
%p A327006 ser := series(exp(1 + (x - 1)*exp(x)), x=0, 25):
%p A327006 seq(n!*coeff(ser, x, n), n = 0..24); # Contributed by C. L. Martin (Retoz), May 16 2021.
%t A327006 With[{m=40}, CoefficientList[Series[Exp[(x-1)*Exp[x] +1], {x,0,m}], x]*Range[0, m]!] (* _G. C. Greubel_, Nov 17 2022 *)
%o A327006 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp((x-1)*Exp(x) + 1) ))); // _G. C. Greubel_, Nov 17 2022
%o A327006 (SageMath)
%o A327006 def A327006_list(prec):
%o A327006     P.<x> = PowerSeriesRing(QQ, prec)
%o A327006     return P( exp((x-1)*exp(x) +1) ).egf_to_ogf().list()
%o A327006 A327006_list(40) # _G. C. Greubel_, Nov 17 2022
%Y A327006 Cf. A005387, A264428, A327005.
%K A327006 nonn
%O A327006 1,4
%A A327006 _Peter Luschny_, Aug 13 2019