A296660 - OEIS

%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