Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Nov 17 2022 07:20:43
%S 1,0,1,2,6,24,105,510,2765,16408,105210,724580,5330149,41649828,
%T 344120777,2995027126,27368953170,261825429024,2615385871053,
%U 27216432127818,294443944669341,3305528914953420,38442535155671262,462431164589185924,5745587267806107545
%N a(n) = A327005(n, n).
%H G. C. Greubel, <a href="/A327006/b327006.txt">Table of n, a(n) for n = 1..540</a>
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%H Retoz, <a href="http://retoz.free.fr/mai2021.pdf">Sous-dérangements</a>, Mai 2021.
%F a(n) = Sum_{k=1..n} BM[n][k] where BM is the BellMatrix(j -> j mod n) as defined in A264428.
%F Assuming offset = 0:
%F a(n) = n! * [x^n] exp(1 + (x - 1)*exp(x)). - Contributed by C. L. Martin (Retoz), May 16 2021.
%F a(n) = Sum_{j=0..n} binomial(n, j) * (-2)^(n-j) * A005387(j). - _G. C. Greubel_, Nov 17 2022
%p # BellMatrix is defined in A264428.
%p a := proc(n) BellMatrix(j -> modp(j, n), n): add(i, i in %[n]) end:
%p seq(a(n), n=1..25);
%p # Or, assuming offset = 0:
%p ser := series(exp(1 + (x - 1)*exp(x)), x=0, 25):
%p seq(n!*coeff(ser, x, n), n = 0..24); # Contributed by C. L. Martin (Retoz), May 16 2021.
%t 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 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp((x-1)*Exp(x) + 1) ))); // _G. C. Greubel_, Nov 17 2022
%o (SageMath)
%o def A327006_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp((x-1)*exp(x) +1) ).egf_to_ogf().list()
%o A327006_list(40) # _G. C. Greubel_, Nov 17 2022
%Y Cf. A005387, A264428, A327005.
%K nonn
%O 1,4
%A _Peter Luschny_, Aug 13 2019