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%I #45 Dec 05 2023 08:59:06
%S 1,5,29,189,1357,10589,88909,797085,7583373,76179037,804638925,
%T 8904557341,102929260813,1239432543709,15511264432973,201330839371421,
%U 2705249923950477,37567754666530141,538369104335121869
%N Row sums of A078939.
%C Equals A078944(n+1)/4.
%H Vincenzo Librandi, <a href="/A078945/b078945.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: exp(4*(exp(x)-1)+x).
%F Stirling transform of [1, 4, 4^2, 4^3, ...]. - _Gerald McGarvey_, Jun 01 2005
%F Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n)=e^{-4}*f_n(4). - _Milan Janjic_, May 30 2008
%F G.f.: 1/(Q(0) - 4*x) where Q(k) = 1 - x*(k+1)/( 1 - 4*x/Q(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Mar 22 2013
%F G.f.: T(0)/(1-5*x), where T(k) = 1 - 4*x^2*(k+1)/( 4*x^2*(k+1) - (1-5*x-x*k)*(1-6*x-x*k)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 28 2013
%F a(n) = exp(-4) * Sum_{k>=0} (k + 1)^n * 4^k / k!. - _Ilya Gutkovskiy_, Apr 20 2020
%F a(n) ~ n^(n+1) * exp(n/LambertW(n/4) - n - 4) / (4 * sqrt(1 + LambertW(n/4)) * LambertW(n/4)^(n+1)). - _Vaclav Kotesovec_, Jun 26 2022
%F a(0) = 1; a(n) = a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Dec 05 2023
%p A078945 := proc(n) local a,b,i;
%p a := [seq(2,i=1..n)]; b := [seq(1,i=1..n)];
%p exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=4,%),66)) end:
%p seq(A078945(n),n=0..18); # _Peter Luschny_, Mar 30 2011
%t Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4+x), {x, 0, 20}], x]
%t Table[1/E^4/4*Sum[m^n/m!*4^m,{m,0,Infinity}],{n,1,20}] (* _Vaclav Kotesovec_, Mar 12 2014 *)
%t Table[BellB[n+1, 4]/4, {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 26 2022 *)
%Y Column k=4 of A335975.
%Y Cf. A078939, A078944, A000110, A035009, A078940.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 18 2002
%E More terms from _Robert G. Wilson v_, Dec 19 2002
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