A293720 - OEIS

%I #19 Jul 12 2024 21:58:20

%S 1,1,9,25,241,1041,10681,60649,658785,4540321,51972841,415198521,

%T 4988808529,44847866545,563683953561,5586645006601,73228719433921,

%U 788319280278849,10747425123292105,124265401483446361,1757874020223846321,21640338257575264081

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

%H Seiichi Manyama, <a href="/A293720/b293720.txt">Table of n, a(n) for n = 0..612</a>

%F a(n) ~ 2^((3*n-1)/2) * exp(-1/32 + sqrt(2*n)/4 - n/2) * n^(n/2). - _Vaclav Kotesovec_, Oct 15 2017

%F a(n) = (-2*i)^n * Hermite(n, i/4). - _G. C. Greubel_, Jul 12 2024

%t CoefficientList[Series[E^(x + 4*x^2), {x,0,30}], x] * Range[0,30]! (* _Vaclav Kotesovec_, Oct 15 2017 *)

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

%o (Magma)

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

%o Coefficients(R!(Laplace( Exp(x+4*x^2) ))); // _G. C. Greubel_, Jul 12 2024

%o (SageMath)

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

%Y Column k=2 of A293724.

%Y Column k=8 of A359762.

%Y 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).

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 15 2017