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A000180
Expansion of E.g.f. exp(-x)/(1-3x).
(Formerly M2063 N0816)
(Formerly M2063 N0816)
8
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
D-finite with recurrence (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, proof via the e.g.f. and arxiv.org:2109.02112
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}]
(* Alternative: *)
Range[0, 19]! CoefficientList[Series[Exp[-x]/(1 - 3 x), {x, 0, 19}], x] (* Vincenzo Librandi, Aug 15 2015 *)
(* Alternative: *)
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) my(x='x+O('x^33)); Vec(serlaplace(exp(-x) / (1-3*x))) \\ Gheorghe Coserea, Aug 06 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Benoit Cloitre, Nov 02 2003
STATUS
approved