OFFSET
0,2
COMMENTS
Also "AIJ" (ordered, indistinct, labeled) transform of 3,3,3,3...
Third row of array A094416 (generalized ordered Bell numbers).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution, arXiv:quant-ph/0303030, J. Phys. A.: Math. Gen 36 (2003) L273.
C. G. Bower, Transforms (2)
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
FORMULA
E.g.f.: 1/(4-3*exp(x)).
a(n) = 3 * A050352(n), n > 0.
a(n) = Sum_{k=0..n} Stirling2(n, k)*(3^k)*k!.
a(n) = (1/4)*Sum_{k>=0} k^n*(3/4)^k. - Karol A. Penson, Jan 25 2002
a(n) = Sum_{k=0..n} A131689(n,k)*3^k. - Philippe Deléham, Nov 03 2008
G.f. A(x)=B(x)/x, where B(x)=x+3*x^2+21*x^3+... = Sum_{n>=1} b(n)*x^n satisfies 4*B(x)-x = 3*B(x/(1-x)), and b(n)=3*Sum_{k=1..n-1} binomial(n-1,k-1)*b(k), b(1)=1. - Vladimir Kruchinin, Jan 27 2011
a(n) = log(4/3)*Integral_{x = 0..inf} (floor(x))^n * (4/3)^(-x) dx. - Peter Bala, Feb 14 2015
a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(0) = 1; a(n) = 3*a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
MAPLE
b:= proc(n, m) option remember;
`if`(n=0, 3^m*m!, m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 04 2021
MATHEMATICA
a[n_] := PolyLog[-n, 3/4]/4; a[0] = 1; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 14 2011 *)
t = 30; Range[0, t]! CoefficientList[Series[1/(4 - 3 Exp[x]), {x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)
PROG
(PARI) a(n)=ceil(polylog(-n, 3/4)/4) \\ Charles R Greathouse IV, Jul 14 2014
(PARI) my(N=25, x='x+O('x^N)); Vec(serlaplace(1/(4 - 3*exp(x)))) \\ Joerg Arndt, Jan 15 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved