A362177 - OEIS

%I #25 Jul 13 2024 02:14:55

%S 1,1,-5,-17,73,481,-1709,-19025,52753,965953,-1882709,-59839889,

%T 64418905,4372890913,-651783677,-367974620369,-309314089439,

%U 35016249465985,66566286588763,-3715188655737617,-11303745326856599,434518893361657441,1858790804545588915

%N Expansion of e.g.f. exp(x * (1-3*x)).

%H Winston de Greef, <a href="/A362177/b362177.txt">Table of n, a(n) for n = 0..628</a>

%F a(n) = a(n-1) - 6*(n-1)*a(n-2) for n > 1.

%F a(n) = n! * Sum_{k=0..floor(n/2)} (-3)^k / (k! * (n-2*k)!).

%F a(n) = (-sqrt(3))^n * Hermite(n, 1/(2*sqrt(3))). - _G. C. Greubel_, Jul 12 2024

%t With[{m=30}, CoefficientList[Series[Exp[x-3*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*(1-3*x))))

%o (Magma)

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

%o Coefficients(R!(Laplace( Exp(x-3*x^2) ))); // _G. C. Greubel_, Jul 12 2024

%o (SageMath)

%o [(-sqrt(3))^n*hermite(n, 1/(2*sqrt(3))) for n in range(31)] # _G. C. Greubel_, Jul 12 2024

%Y Column k=6 of A362277.

%Y Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), this sequence (q=-3), A362176 (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