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

%I A293720 #19 Jul 12 2024 21:58:20
%S A293720 1,1,9,25,241,1041,10681,60649,658785,4540321,51972841,415198521,
%T A293720 4988808529,44847866545,563683953561,5586645006601,73228719433921,
%U A293720 788319280278849,10747425123292105,124265401483446361,1757874020223846321,21640338257575264081
%N A293720 Expansion of e.g.f.: exp(x + 4*x^2).
%H A293720 Seiichi Manyama, <a href="/A293720/b293720.txt">Table of n, a(n) for n = 0..612</a>
%F A293720 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 A293720 a(n) = (-2*i)^n * Hermite(n, i/4). - _G. C. Greubel_, Jul 12 2024
%t A293720 CoefficientList[Series[E^(x + 4*x^2), {x,0,30}], x] * Range[0,30]! (* _Vaclav Kotesovec_, Oct 15 2017 *)
%o A293720 (PARI) my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x+4*x^2)))
%o A293720 (Magma)
%o A293720 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A293720 Coefficients(R!(Laplace( Exp(x+4*x^2) ))); // _G. C. Greubel_, Jul 12 2024
%o A293720 (SageMath)
%o A293720 [(-2*i)^n*hermite(n, i/4) for n in range(31)] # _G. C. Greubel_, Jul 12 2024
%Y A293720 Column k=2 of A293724.
%Y A293720 Column k=8 of A359762.
%Y A293720 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 A293720 nonn
%O A293720 0,3
%A A293720 _Seiichi Manyama_, Oct 15 2017