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%I #51 Dec 05 2023 08:57:59
%S 1,4,19,103,622,4117,29521,227290,1865881,16239523,149142952,
%T 1439618143,14555631781,153700654036,1690684883191,19328770917499,
%U 229203640111870,2814018686591089,35711716110387589,467766675528462562
%N Row sums of A078938.
%C Divide by 3^n and insert an initial 1 to get sequence that shifts left one place under 1/3 order binomial transformation. - _Franklin T. Adams-Watters_, Jul 13 2006
%C Binomial transform of A027710. - _Vaclav Kotesovec_, Jun 26 2022
%H Vincenzo Librandi, <a href="/A078940/b078940.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: exp(3*(exp(x)-1)+x).
%F Stirling transform of [1, 3, 3^2, 3^3, ...]. - _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^{-3}*f_n(3). - _Milan Janjic_, May 30 2008
%F G.f.: 1/T(0), where T(k) = 1 - (k+4)*x - 3*(k+1)*x^2/T(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 15 2016
%F a(n) = exp(-3) * Sum_{k>=0} (k + 1)^n * 3^k / k!. - _Ilya Gutkovskiy_, Apr 20 2020
%F a(n) ~ n^(n+1) * exp(n/LambertW(n/3) - n - 3) / (3 * sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n+1)). - _Vaclav Kotesovec_, Jun 26 2022
%F a(0) = 1; a(n) = a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Dec 05 2023
%p A078940 := 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=3,%),66)) end:
%p seq(A078940(n),n=0..19); # _Peter Luschny_, Mar 30 2011
%t Table[n!, {n, 0, 20}]CoefficientList[Series[E^(3E^x-3+x), {x, 0, 20}], x]
%t Table[1/E^3/3*Sum[m^n/m!*3^m,{m,0,Infinity}],{n,1,20}] (* _Vaclav Kotesovec_, Mar 12 2014 *)
%t Table[BellB[n+1, 3]/3, {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 15 2016 *)
%t nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - (k+4)*x - 3*(k+1)*x^2/g[k+1]; CoefficientList[Series[1/g[0], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 15 2016, after _Sergei N. Gladkovskii_ *)
%Y Column k=3 of A335975.
%Y Cf. A027710, A035009, A078938, A078945, A355254.
%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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