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%I M2063 N0816 #65 May 06 2022 16:34:09
%S 1,2,13,116,1393,20894,376093,7897952,189550849,5117872922,
%T 153536187661,5066694192812,182400990941233,7113638646708086,
%U 298772823161739613,13444777042278282584,645349298029357564033,32912814199497235765682
%N Expansion of E.g.f. exp(-x)/(1-3x).
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000180/b000180.txt">Table of n, a(n) for n=0..100</a>
%F a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*k!*3^k. - _Benoit Cloitre_, Nov 02 2003
%F a(n) = {(3^n*n!)/exp(1/3)}, where {x} = nearest integer. - _Simon Plouffe_, Feb 17 2011
%F Conjecture: (n+1)*a(n) -(n+1)*(3*n-1)*a(n-1) -3*(n-1)*(n+1)*a(n-2) = 0. - _R. J. Mathar_, Jul 24 2012
%F E.g.f.: exp(-x)/(1-3x) = A(x) satisfies (1-3x)A' - (2+3x)A = 0. - _Gheorghe Coserea_, Aug 06 2015
%F a(n+1) = (3*n+2)*a(n) + 3*n*a(n-1). - _Gheorghe Coserea_, Aug 06 2015
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (3*k - 1) * a(n-k). - _Ilya Gutkovskiy_, Jan 17 2020
%F a(n) = 3*n*a(n-1)+(-1)^n for n > 0. - _Werner Schulte_, Mar 09 2020
%t FunctionExpand @ Table[ Gamma[n, -1/3]*3^(n-1)/Exp[ 1/3 ], {n, 24}]
%t Range[0, 19]! CoefficientList[Series[Exp[-x]/(1 - 3 x), {x, 0, 19}], x] (* _Vincenzo Librandi_, Aug 15 2015 *)
%t a[n_] := 3^n n! Sum[(-1)^i/(3^i i!), {i, 0, n}]; Table[a[n], {n, 0, 20}] (* _Gerry Martens_ , May 06 2016 *)
%o (PARI) x='x+O('x^33); Vec(serlaplace(exp(-x) / (1-3*x))) \\ _Gheorghe Coserea_, Aug 06 2015
%Y Column k=3 of A320032.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E More terms from _Benoit Cloitre_, Nov 02 2003
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