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

Author

Paul Barry, Aug 26 2004

Keywords

• easy
• nonn

Comments

Second binomial transform of n!4^n.

Links

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

Crossrefs

Cf. A000165, A010845.

Programs

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 *)

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

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)