OFFSET
0,2
COMMENTS
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
Binomial transform of A027710. - Vaclav Kotesovec, Jun 26 2022
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f.: exp(3*(exp(x)-1)+x).
Stirling transform of [1, 3, 3^2, 3^3, ...]. - Gerald McGarvey, Jun 01 2005
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
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
a(n) = exp(-3) * Sum_{k>=0} (k + 1)^n * 3^k / k!. - Ilya Gutkovskiy, Apr 20 2020
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
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
MAPLE
A078940 := proc(n) local a, b, i;
a := [seq(2, i=1..n)]; b := [seq(1, i=1..n)];
exp(-x)*hypergeom(a, b, x); round(evalf(subs(x=3, %), 66)) end:
seq(A078940(n), n=0..19); # Peter Luschny, Mar 30 2011
MATHEMATICA
Table[n!, {n, 0, 20}]CoefficientList[Series[E^(3E^x-3+x), {x, 0, 20}], x]
Table[1/E^3/3*Sum[m^n/m!*3^m, {m, 0, Infinity}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 3]/3, {n, 0, 20}] (* Vaclav Kotesovec, Jan 15 2016 *)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 18 2002
EXTENSIONS
More terms from Robert G. Wilson v, Dec 19 2002
STATUS
approved