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

1, 1, -3, -11, 25, 201, -299, -5123, 3249, 167185, 50221, -6637179, -8846903, 309737689, 769776645, -16575533939, -62762132639, 998072039457, 5265897058909, -66595289781995, -466803466259079, 4860819716300521, 44072310882063157, -383679824152382691

Offset: 0

Author

Seiichi Manyama, Apr 10 2023

Keywords

• sign
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..668

Crossrefs

Column k=4 of A362277.

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).

Programs

Magma

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

Mathematica

With[{m=30}, CoefficientList[Series[Exp[x-2*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*(1-2*x))))

SageMath

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

Formula

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

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

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