OFFSET
0,2
COMMENTS
For further information, references, programs, etc. for r-Bell numbers see A005493. - N. J. A. Sloane, Nov 27 2013
From expansion of falling factorials (binomial transform of A005493).
Row sums of Sheffer triangle (exp(3*x), exp(x)-1). - Wolfdieter Lang, Sep 29 2011
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.
J. Riordan, Letter, Oct 31 1977
N. J. A. Sloane, Transforms
Earl Glen Whitehead Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.
FORMULA
a(n) = Sum_{i=0..n} 3^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007
a(n) = exp(-1)*Sum_{k>=0} ((k+3)^n)/k!. - Gerald McGarvey, Jun 03 2004. May be rewritten as a(n) = Sum_{k>=3} (k^n*(k-1)*(k-2)/k!)/exp(1), which is a Dobinski-type relation for this sequence. - Karol A. Penson, Aug 18 2006
Define f_1(x), f_2(x), ... such that f_1(x) = x^2*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i <= j), and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = (-1)^(n)charpoly(A,-3). - Milan Janjic, Jul 08 2010
a(n) = Sum_{k=3..n+3} A143495(n+3,k), n >= 0. - Wolfdieter Lang, Sep 29 2011
G.f.: 1/U(0) where U(k)= 1 - x*(k+4) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: Sum_{k>0} x^(k-1) / ((1 - 3*x) * (1 - 4*x) * ... * (1 - (k+2)*x)). - Michael Somos, Feb 26 2014
G.f.: Sum_{k>0} k * x^(k-1) / ((1 - 2*x) * (1 - 3*x) * ... * (1 - (k+1)*x)). - Michael Somos, Feb 26 2014
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 3) / LambertW(n)^(n + 7/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{k=0..n} 4^k*A124323(n, k). - Mélika Tebni, Jun 10 2022
EXAMPLE
G.f. = 1 + 4*x + 17*x^2 + 77*x^3 + 372*x^4 + 1915*x^5 + 10481*x^6 + 60814*x^7 + ...
MAPLE
seq(add(3^(n-i)*combinat:-bell(i)*binomial(n, i), i=0..n), n=0..50); # Robert Israel, Dec 16 2014
MATHEMATICA
Range[0, 40]! CoefficientList[Series[Exp[3 x + Exp[x] - 1], {x, 0, 40}], x] (* Vincenzo Librandi, Mar 04 2014 *)
PROG
(Magma)
A005494:= func< n | (&+[Binomial(n, j)*3^(n-j)*Bell(j): j in [0..n]]) >;
[A005494(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022
(SageMath)
def A005494(n): return sum( 3^(n-j)*bell_number(j)*binomial(n, j) for j in range(n+1))
[A005494(n) for n in range(31)] # G. C. Greubel, Dec 01 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved