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A362176 Expansion of e.g.f. exp(x * (1-2*x)).

%I A362176 #18 Jul 12 2024 21:58:00
%S A362176 1,1,-3,-11,25,201,-299,-5123,3249,167185,50221,-6637179,-8846903,
%T A362176 309737689,769776645,-16575533939,-62762132639,998072039457,
%U A362176 5265897058909,-66595289781995,-466803466259079,4860819716300521,44072310882063157,-383679824152382691
%N A362176 Expansion of e.g.f. exp(x * (1-2*x)).
%H A362176 Seiichi Manyama, <a href="/A362176/b362176.txt">Table of n, a(n) for n = 0..668</a>
%F A362176 a(n) = a(n-1) - 4*(n-1)*a(n-2) for n > 1.
%F A362176 a(n) = n! * Sum_{k=0..floor(n/2)} (-2)^k / (k! * (n-2*k)!).
%F A362176 a(n) = (-sqrt(2))^n * Hermite(n, 1/(2*sqrt(2))). - _G. C. Greubel_, Jul 12 2024
%t A362176 With[{m=30}, CoefficientList[Series[Exp[x-2*x^2], {x,0,m}], x]*Range[0, m]!] (* _G. C. Greubel_, Jul 12 2024 *)
%o A362176 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-2*x))))
%o A362176 (Magma)
%o A362176 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A362176 Coefficients(R!(Laplace( Exp(x-2*x^2) ))); // _G. C. Greubel_, Jul 12 2024
%o A362176 (SageMath)
%o A362176 [(-sqrt(2))^n*hermite(n, 1/(2*sqrt(2))) for n in range(31)] # _G. C. Greubel_, Jul 12 2024
%Y A362176 Column k=4 of A362277.
%Y A362176 Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), this sequence (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).
%K A362176 sign,easy
%O A362176 0,3
%A A362176 _Seiichi Manyama_, Apr 10 2023