%I #44 Mar 02 2017 06:05:11

%S 1,4,25,226,2713,40696,732529,15383110,369194641,9968255308,

%T 299047659241,9868572754954,355268619178345,13855476147955456,

%U 581929998214129153,26186849919635811886,1256968796142518970529

%N a(n) = 3*n*a(n-1) + 1, a(0) = 1.

%C a(n)/(A000142*A000244) is an increasingly good approximation to cube root of e.

%C Related to Incomplete Gamma Function at 1/3. - _Michael Somos_, Mar 26 1999

%C For positive n, a(n) equals 3^n times the permanent of the n X n matrix with (4/3)'s along the main diagonal, and 1's everywhere else. - _John M. Campbell_, Jul 10 2011

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.

%H Vincenzo Librandi, <a href="/A010845/b010845.txt">Table of n, a(n) for n = 0..200</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.

%H Roland Bacher, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p7">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013

%H M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html"> The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

%F E.g.f.: exp(x)/(1-3*x).

%F a(n) = floor( n!*e^(1/3)*3^n ) = n! * (Sum_{k=0..n} 3^(n-k) / k!) = n! * (e^(1/3) * 3^n - Sum_{k>n} 3^(n-k) / k!). - _Michael Somos_, Mar 26 1999

%F a(n) = Sum_{k=0..n} P(n, k)*3^k. - _Ross La Haye_, Aug 29 2005

%F Binomial transform of A032031. - _Carl Najafi_, Sep 11 2011

%F Conjecture: a(n) +(-3*n-1)*a(n-1) +3*(n-1)*a(n-2)=0. - _R. J. Mathar_, Feb 16 2014

%F a(n) = hypergeometric_U(1,n+2,1/3)/3. - _Peter Luschny_, Nov 26 2014

%F From _Peter Bala_, Mar 01 2017: (Start)

%F a(n) = Integral_{x = 0..inf} (3*x + 1)^n*exp(-x) dx.

%F The e.g.f. y = exp(x)/(1 - 3*x) satisfies the differential equation (1 - 3*x)*y' = (4 - 3*x)*y. Mathar's recurrence above follows easily from this.

%F The sequence b(n) := 3^n*n! also satisfies Mathar's recurrence with b(0) = 1, b(1) = 3. This leads to the continued fraction representation a(n) = 3^n*n!*( 1 + 1/(3 - 3/(7 - 6/(10 - ... - (3*n - 3)/(3*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/3) = 1 + 1/(3 - 3/(7 - 6/(10 - ... - (3*n - 2)/((3*n + 1) - ... )))). Cf. A010844. (End)

%e 1 + 4*x + 25*x^2 + 226*x^3 + 2713*x^4 + 40696*x^5 + 732529*x^6 + ...

%t Table[ Gamma[ n, 1/3 ]*Exp[ 1/3 ]*3^(n-1), {n, 1, 24} ]

%t a[ n_] := If[ n<0, 0, Floor[ n! E^(1/3) 3^n ]] (* _Michael Somos_, Sep 04 2013 *)

%t Range[0, 20]! CoefficientList[Series[Exp[x]/(1 - 3 x), {x, 0, 20}], x] (* _Vincenzo Librandi_, Feb 17 2014 *)

%o (PARI) {a(n) = if( n<0, 0, n! * sum(k=0, n, 3^(n-k) / k!))} /* _Michael Somos_, Sep 04 2013 */

%Y Cf. A000522, A010844, A056545, A056546, A056547 for analogs.

%K nonn,easy

%O 0,2

%A _Simon Plouffe_

%E Better description and formulas from _Michael Somos_

%E More terms from _James A. Sellers_, Jul 04 2000