A097820 - OEIS

%I #26 Dec 14 2020 12:58:27

%S 1,6,52,632,10128,202592,4862272,136143744,4356600064,156837602816,

%T 6273504113664,276034181003264,13249640688160768,688981315784368128,

%U 38582953683924631552,2314977221035477925888,148158542146270587322368

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

%C Second binomial transform of n!4^n.

%H Michael De Vlieger, <a href="/A097820/b097820.txt">Table of n, a(n) for n = 0..365</a>

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

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

%F a(n) +2*(-2*n-1)*a(n-1) +8*(n-1)*a(n-2) = 0. - _R. J. Mathar_, Feb 19 2015

%F From _Emanuele Munarini_, Dec 18 2017: (Start)

%F a(n) = Sum_{k=0..n} binomial(n,k)*4^k*k!*2^(n-k).

%F Sum_{k=0..n} binomial(n,k)*(-2)^(n-k)*a(k) = 4^n*n!. (End)

%F From _Vaclav Kotesovec_, Dec 18 2017: (Start)

%F a(n) = exp(1/2) * 4^n * Gamma(n + 1, 1/2).

%F a(n) ~ n! * exp(1/2) * 4^n. (End)

%p 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):

%p map(f, [$0..50]); # _Robert Israel_, Dec 19 2017

%t Table[Sum[Binomial[n,k]4^k k! 2^(n-k),{k,0,n}],{n,0,12}] (* _Emanuele Munarini_, Dec 18 2017 *)

%t Fold[Append[#1, 4 #2*#1[[#2]] + 2^#2] &, {1}, Range@ 16] (* _Michael De Vlieger_, Dec 18 2017 *)

%t With[{nn=20},CoefficientList[Series[Exp[2x]/(1-4x),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Dec 14 2020 *)

%o (Maxima) makelist(sum(binomial(n,k)*4^k*k!*2^(n-k),k,0,n),n,0,12); /* _Emanuele Munarini_, Dec 18 2017 */

%o (PARI) x='x+O('x^99); Vec(serlaplace(exp(2*x)/(1-4*x))) \\ _Altug Alkan_, Dec 18 2017

%Y Cf. A000165, A010845.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Aug 26 2004