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Expansion of e.g.f. exp(x^2) / (1 - x * exp(x^2/2)).
3

%I #9 Aug 23 2024 07:29:19

%S 1,1,4,15,84,555,4440,41265,438480,5240025,69582240,1016350335,

%T 16194911040,279560396115,5197054262400,103514720133825,

%U 2199255573715200,49645309340451825,1186599954328588800,29937224154635772975,795051251297099596800

%N Expansion of e.g.f. exp(x^2) / (1 - x * exp(x^2/2)).

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

%p A375633 := proc(n)

%p n!*add(((n-2*k+2)/2)^k/k!,k=0..floor(n/2)) ;

%p end proc:

%p seq(A375633(n),n=0..60) ; # _R. J. Mathar_, Aug 23 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2)/(1-x*exp(x^2/2))))

%o (PARI) a(n) = n!*sum(k=0, n\2, ((n-2*k+2)/2)^k/k!);

%Y Cf. A375394, A375627.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 22 2024