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%I #19 Mar 01 2024 02:05:55
%S 1,2,20,232,3728,74528,1788736,50084480,1602703616,57697329664,
%T 2307893187584,101547300251648,4874270412083200,253462061428318208,
%U 14193875439985836032,851632526399150129152,54504481689545608331264,3706304754889101366394880
%N Expansion of the e.g.f. exp(-2*x)/(1-4*x).
%C Binomial self-convolution of sequence A296618.
%F E.g.f.: exp(-2*x)/(1-4*x).
%F a(n) = Sum_{k=0..n} binomial(n,k)*4^k*k!*(-2)^(n-k).
%F Sum_{k=0..n} binomial(n,k)*2^(n-k)*a(k) = 4^n n!.
%F a(n+1)-4*(n+1)*a(n) = (-2)^(n+1).
%F D-finite with recurrence a(n+2)-(4*n+6)*a(n+1)-8*(n+1)*a(n) = 0.
%F From _Vaclav Kotesovec_, Dec 18 2017: (Start)
%F a(n) = exp(-1/2) * 4^n * Gamma(n + 1, -1/2).
%F a(n) ~ n! * exp(-1/2) * 4^n. (End)
%t CoefficientList[Series[Exp[-2x]/(1-4x),{x,0,12}],x]Range[0,12]!
%t Table[Sum[Binomial[n, k] 4^k k! (-2)^(n-k), {k, 0, n}], {n, 0, 12}]
%o (Maxima) makelist(sum(binomial(n,k)*4^k*k!*(-2)^(n-k),k,0,n),n,0,12);
%o (PARI) x='x+O('x^99); Vec(serlaplace(exp(-2*x)/(1-4*x))) \\ _Altug Alkan_, Dec 18 2017
%Y Cf. A001907, A056545, A097820, A296618.
%K nonn
%O 0,2
%A _Emanuele Munarini_, Dec 18 2017
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