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

1, 1, -1, -5, 1, 41, 31, -461, -895, 6481, 22591, -107029, -604031, 1964665, 17669471, -37341149, -567425279, 627491489, 19919950975, -2669742629, -759627879679, -652838174519, 31251532771999, 59976412450835, -1377594095061119, -4256461892701199

Offset: 0

Author

Seiichi Manyama, Oct 12 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..732

Crossrefs

Cf. A000321, A111884, A293571, A293572, A293573.

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).

Programs

Magma

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

Mathematica

CoefficientList[Series[E^(x*(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)

PARI

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x))))

PARI

a(n) = polhermite(n, 1/2); \\ Michel Marcus, Oct 13 2017

SageMath

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

Formula

a(n) = (-1)^n * A000321(n).

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

E.g.f.: Product_{k>=1} (1 + x^k)^(mu(k)/k). - Ilya Gutkovskiy, May 23 2019

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