A362177 Expansion of e.g.f. exp(x * (1-3*x)).

1, 1, -5, -17, 73, 481, -1709, -19025, 52753, 965953, -1882709, -59839889, 64418905, 4372890913, -651783677, -367974620369, -309314089439, 35016249465985, 66566286588763, -3715188655737617, -11303745326856599, 434518893361657441, 1858790804545588915

Offset: 0

Author

Seiichi Manyama, Apr 10 2023

Keywords

• sign
• easy

Links

• Winston de Greef, Table of n, a(n) for n = 0..628

Crossrefs

Column k=6 of A362277.

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).

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-3*x^2) ))); // G. C. Greubel, Jul 12 2024

Mathematica

With[{m=30}, CoefficientList[Series[Exp[x-3*x^2], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Jul 12 2024 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-3*x))))

SageMath

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

Formula

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

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

a(n) = (-sqrt(3))^n * Hermite(n, 1/(2*sqrt(3))). - G. C. Greubel, Jul 12 2024