login
Expansion of e.g.f. exp(x * (1-2*x)).
10

%I #18 Jul 12 2024 21:58:00

%S 1,1,-3,-11,25,201,-299,-5123,3249,167185,50221,-6637179,-8846903,

%T 309737689,769776645,-16575533939,-62762132639,998072039457,

%U 5265897058909,-66595289781995,-466803466259079,4860819716300521,44072310882063157,-383679824152382691

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

%H Seiichi Manyama, <a href="/A362176/b362176.txt">Table of n, a(n) for n = 0..668</a>

%F a(n) = a(n-1) - 4*(n-1)*a(n-2) for n > 1.

%F a(n) = n! * Sum_{k=0..floor(n/2)} (-2)^k / (k! * (n-2*k)!).

%F a(n) = (-sqrt(2))^n * Hermite(n, 1/(2*sqrt(2))). - _G. C. Greubel_, Jul 12 2024

%t With[{m=30}, CoefficientList[Series[Exp[x-2*x^2], {x,0,m}], x]*Range[0, m]!] (* _G. C. Greubel_, Jul 12 2024 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-2*x))))

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o Coefficients(R!(Laplace( Exp(x-2*x^2) ))); // _G. C. Greubel_, Jul 12 2024

%o (SageMath)

%o [(-sqrt(2))^n*hermite(n, 1/(2*sqrt(2))) for n in range(31)] # _G. C. Greubel_, Jul 12 2024

%Y Column k=4 of A362277.

%Y 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 sign,easy

%O 0,3

%A _Seiichi Manyama_, Apr 10 2023