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%I #17 Aug 20 2021 05:25:36
%S 1,0,0,4,12,48,400,3120,25872,251776,2715264,31809600,405296320,
%T 5580385536,82469607168,1302102360832,21875297337600,389590168842240,
%U 7331376554610688,145352459953603584,3028176414606560256,66135374473635328000,1510938930307368898560,36038691473858577444864
%N E.g.f.: exp(-x * (2 + x)) / (1 - x)^2.
%F a(0) = 1; a(n) = 2 * Sum_{k=3..n} binomial(n-1,k-1) * (k-1)! * a(n-k).
%F a(n) = Sum_{k=0..n} binomial(n,k) * Hermite(k,-1) * (n-k+1)!.
%F a(n) = Sum_{k=0..n} binomial(n,k) * A038205(k) * A038205(n-k).
%F a(n) ~ exp(-3) * n * n!. - _Vaclav Kotesovec_, Aug 09 2021
%F D-finite with recurrence a(n) +(-n+1)*a(n-1) -2*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Aug 20 2021
%p A335595 := proc(n)
%p option remember ;
%p if n = 0 then
%p 1;
%p else
%p 2*add(binomial(n-1,k-1)*(k-1)!*procname(n-k),k=3..n) ;
%p end if;
%p end proc:
%p seq(A335595(n),n=0..42) ; # _R. J. Mathar_, Aug 20 2021
%t nmax = 23; CoefficientList[Series[Exp[-x (2 + x)]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] (k - 1)! a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 23}]
%t Table[Sum[Binomial[n, k] HermiteH[k, -1] (n - k + 1)!, {k, 0, n}], {n, 0, 23}]
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(-x*(2+x))/(1-x)^2)) \\ _Michel Marcus_, Nov 21 2020
%Y Cf. A038205, A087981.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Nov 20 2020
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