A293720 Expansion of e.g.f.: exp(x + 4*x^2).

1, 1, 9, 25, 241, 1041, 10681, 60649, 658785, 4540321, 51972841, 415198521, 4988808529, 44847866545, 563683953561, 5586645006601, 73228719433921, 788319280278849, 10747425123292105, 124265401483446361, 1757874020223846321, 21640338257575264081

Offset: 0

Author

Seiichi Manyama, Oct 15 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..612

Crossrefs

Column k=2 of A293724.

Column k=8 of A359762.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), this sequence (q=4).

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x+4*x^2) ))); // G. C. Greubel, Jul 12 2024

Mathematica

CoefficientList[Series[E^(x + 4*x^2), {x,0,30}], x] * Range[0,30]! (* Vaclav Kotesovec, Oct 15 2017 *)

PARI

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x+4*x^2)))

SageMath

[(-2*i)^n*hermite(n, i/4) for n in range(31)] # G. C. Greubel, Jul 12 2024

Formula

a(n) ~ 2^((3*n-1)/2) * exp(-1/32 + sqrt(2*n)/4 - n/2) * n^(n/2). - Vaclav Kotesovec, Oct 15 2017

a(n) = (-2*i)^n * Hermite(n, i/4). - G. C. Greubel, Jul 12 2024