A000180 Expansion of E.g.f. exp(-x)/(1-3x). (Formerly M2063 N0816)

1, 2, 13, 116, 1393, 20894, 376093, 7897952, 189550849, 5117872922, 153536187661, 5066694192812, 182400990941233, 7113638646708086, 298772823161739613, 13444777042278282584, 645349298029357564033, 32912814199497235765682

Offset

0,2

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=0..100

Formula

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*k!*3^k. - Benoit Cloitre, Nov 02 2003

a(n) = {(3^n*n!)/exp(1/3)}, where {x} = nearest integer. - Simon Plouffe, Feb 17 2011

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

E.g.f.: exp(-x)/(1-3x) = A(x) satisfies (1-3x)A' - (2+3x)A = 0. - Gheorghe Coserea, Aug 06 2015

a(n+1) = (3*n+2)*a(n) + 3*n*a(n-1). - Gheorghe Coserea, Aug 06 2015

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (3*k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020

a(n) = 3*n*a(n-1)+(-1)^n for n > 0. - Werner Schulte, Mar 09 2020

Mathematica

FunctionExpand @ Table[ Gamma[n, -1/3]*3^(n-1)/Exp[ 1/3 ], {n, 24}]
Range[0, 19]! CoefficientList[Series[Exp[-x]/(1 - 3 x), {x, 0, 19}], x] (* Vincenzo Librandi, Aug 15 2015 *)
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 *)

Prog

(PARI) x='x+O('x^33); Vec(serlaplace(exp(-x) / (1-3*x))) \\ Gheorghe Coserea, Aug 06 2015

Crossrefs

Column k=3 of A320032.

Sequence in context: A208958 A209052 A209217 * A215715 A292437 A317196

Adjacent sequences: A000177 A000178 A000179 * A000181 A000182 A000183

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Benoit Cloitre, Nov 02 2003

Status

approved