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A293604 Expansion of e.g.f.: exp(x * (1 - x)).

%I A293604 #22 Jul 12 2024 21:59:55
%S A293604 1,1,-1,-5,1,41,31,-461,-895,6481,22591,-107029,-604031,1964665,
%T A293604 17669471,-37341149,-567425279,627491489,19919950975,-2669742629,
%U A293604 -759627879679,-652838174519,31251532771999,59976412450835,-1377594095061119,-4256461892701199
%N A293604 Expansion of e.g.f.: exp(x * (1 - x)).
%H A293604 Seiichi Manyama, <a href="/A293604/b293604.txt">Table of n, a(n) for n = 0..732</a>
%F A293604 a(n) = (-1)^n * A000321(n).
%F A293604 a(n) = a(n-1) - 2 * (n-1) * a(n-2) for n > 1.
%F A293604 E.g.f.: Product_{k>=1} (1 + x^k)^(mu(k)/k). - _Ilya Gutkovskiy_, May 23 2019
%F A293604 a(n) = Hermite(n, 1/2). - _G. C. Greubel_, Jul 12 2024
%t A293604 CoefficientList[Series[E^(x*(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Oct 13 2017 *)
%o A293604 (PARI) my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x))))
%o A293604 (PARI) a(n) = polhermite(n, 1/2); \\ _Michel Marcus_, Oct 13 2017
%o A293604 (Magma)
%o A293604 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A293604 Coefficients(R!(Laplace( Exp(x-x^2) ))); // _G. C. Greubel_, Jul 12 2024
%o A293604 (SageMath)
%o A293604 [hermite(n, 1/2) for n in range(31)] # _G. C. Greubel_, Jul 12 2024
%Y A293604 Cf. A000321, A111884, A293571, A293572, A293573.
%Y A293604 Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), this sequence (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).
%K A293604 sign
%O A293604 0,4
%A A293604 _Seiichi Manyama_, Oct 12 2017