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

0,2

Comments

Binomial self-convolution of sequence A296618.

Links

Table of n, a(n) for n=0..17.

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

Cf. A001907, A056545, A097820, A296618.

Sequence in context: A127110 A361964 A337856 * A197898 A293471 A109106

Adjacent sequences: A296657 A296658 A296659 * A296661 A296662 A296663

Keyword

nonn

Author

Emanuele Munarini, Dec 18 2017

Status

approved