OFFSET
0,2
COMMENTS
Binomial self-convolution of sequence A296618.
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)
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}]
PROG
(Maxima) makelist(sum(binomial(n, k)*4^k*k!*(-2)^(n-k), k, 0, n), n, 0, 12);
(PARI) x='x+O('x^99); Vec(serlaplace(exp(-2*x)/(1-4*x))) \\ Altug Alkan, Dec 18 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Dec 18 2017
STATUS
approved