A296660 Expansion of the e.g.f. exp(-2*x)/(1-4*x).
1, 2, 20, 232, 3728, 74528, 1788736, 50084480, 1602703616, 57697329664, 2307893187584, 101547300251648, 4874270412083200, 253462061428318208, 14193875439985836032, 851632526399150129152, 54504481689545608331264, 3706304754889101366394880
Offset: 0
Keywords
Programs
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Mathematica
CoefficientList[Series[Exp[-2x]/(1-4x),{x,0,12}],x]Range[0,12]! Table[Sum[Binomial[n, k] 4^k k! (-2)^(n-k), {k, 0, n}], {n, 0, 12}] -
Maxima
makelist(sum(binomial(n,k)*4^k*k!*(-2)^(n-k),k,0,n),n,0,12);
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PARI
x='x+O('x^99); Vec(serlaplace(exp(-2*x)/(1-4*x))) \\ Altug Alkan, Dec 18 2017
Formula
E.g.f.: exp(-2*x)/(1-4*x).
a(n) = Sum_{k=0..n} binomial(n,k)*4^k*k!*(-2)^(n-k).
Sum_{k=0..n} binomial(n,k)*2^(n-k)*a(k) = 4^n n!.
a(n+1)-4*(n+1)*a(n) = (-2)^(n+1).
D-finite with recurrence a(n+2)-(4*n+6)*a(n+1)-8*(n+1)*a(n) = 0.
From Vaclav Kotesovec, Dec 18 2017: (Start)
a(n) = exp(-1/2) * 4^n * Gamma(n + 1, -1/2).
a(n) ~ n! * exp(-1/2) * 4^n. (End)
Comments