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%I #11 Jul 28 2020 11:08:06
%S 1,1,3,17,137,1389,16819,236557,3792753,68326073,1366917731,
%T 30074632521,721798881913,18766625660197,525460685327187,
%U 15763716503597189,504436925448024929,17150818356045629937,617428780939911647683,23462281235407345160833
%N E.g.f.: exp(-x/(1 - x)) / (1 - 2*x).
%H Robert Israel, <a href="/A331688/b331688.txt">Table of n, a(n) for n = 0..403</a>
%F a(n) = Sum_{k=0..n} binomial(n,k)^2 * k! * A000166(n-k).
%F a(n) = Sum_{k=0..n} binomial(n,k) * k! * 2^k * A293116(n-k).
%F a(n) ~ n! * exp(-1) * 2^n. - _Vaclav Kotesovec_, Jan 26 2020
%F a(n) = (4*n-3)*a(n-1)-(n-1)*(5*n-8)*a(n-2)+2*(n-1)*(n--2)^2*a(n-3). - _Robert Israel_, Jul 28 2020
%p f:= gfun:-rectoproc({a(n) = -(n - 1)*(5*n - 8)*a(n - 2) + (-3 + 4*n)*a(n - 1) + 2*(n - 1)*(n - 2)^2*a(n - 3),a(0)=1,a(1)=1,a(2)=3},a(n),remember):
%p map(f, [$0..30]); # _Robert Israel_, Jul 28 2020
%t nmax = 19; CoefficientList[Series[Exp[-x/(1 - x)]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[Binomial[n, k]^2 k! Subfactorial[n - k], {k, 0, n}], {n, 0, 19}]
%Y Cf. A000165, A000166, A000354, A009940, A293116, A331689.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 24 2020
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