A097820 Expansion of e.g.f. exp(2*x)/(1-4*x).

1, 6, 52, 632, 10128, 202592, 4862272, 136143744, 4356600064, 156837602816, 6273504113664, 276034181003264, 13249640688160768, 688981315784368128, 38582953683924631552, 2314977221035477925888, 148158542146270587322368

Offset

0,2

Comments

Second binomial transform of n!4^n.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..365

Formula

E.g.f.: exp(2x)/(1-4x).

a(n) = 4n*a(n-1)+2^n, n>0, a(0)=1.

a(n) +2*(-2*n-1)*a(n-1) +8*(n-1)*a(n-2) = 0. - R. J. Mathar, Feb 19 2015

From Emanuele Munarini, Dec 18 2017: (Start)

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!. (End)

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)

Maple

f:= rectoproc({a(n) +2*(-2*n-1)*a(n-1) +8*(n-1)*a(n-2) = 0, a(0)=1, a(1)=6}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Dec 19 2017

Mathematica

Table[Sum[Binomial[n, k]4^k k! 2^(n-k), {k, 0, n}], {n, 0, 12}] (* Emanuele Munarini, Dec 18 2017 *)
Fold[Append[#1, 4 #2*#1[[#2]] + 2^#2] &, {1}, Range@ 16] (* Michael De Vlieger, Dec 18 2017 *)
With[{nn=20}, CoefficientList[Series[Exp[2x]/(1-4x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Dec 14 2020 *)

Prog

(Maxima) makelist(sum(binomial(n, k)*4^k*k!*2^(n-k), k, 0, n), n, 0, 12); /* Emanuele Munarini, Dec 18 2017 */
(PARI) x='x+O('x^99); Vec(serlaplace(exp(2*x)/(1-4*x))) \\ Altug Alkan, Dec 18 2017

Crossrefs

Cf. A000165, A010845.

Sequence in context: A294158 A209306 A271802 * A166889 A164894 A027835

Adjacent sequences: A097817 A097818 A097819 * A097821 A097822 A097823

Keyword

easy,nonn

Author

Paul Barry, Aug 26 2004

Status

approved