OFFSET
0,2
COMMENTS
Also, number of preferential fuzzy subsets of length n+2 where the keychains are of length n+2.
Binomial transform of A007047.
Double binomial transform of A000629.
Conjecture: for fixed k = 1,2,..., the sequence obtained by reducing a(n) modulo k is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 24 the sequence becomes [1, 4, 18, 22, 6, 22, 6, 22, 6, ...] with an apparent period of 2 beginning at a(3). - Peter Bala, Jul 08 2022
LINKS
Robert Israel, Table of n, a(n) for n = 0..387
V. Murali, Ordered partitions and finite fuzzy sets, Far East J. Math. Sci.(FJMS), 21(2006), 12-132.
S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO], 2015.
FORMULA
E.g.f.: exp(3*x)/(2-exp(x)).
a(n) = 3^n + Sum_{k = 0..n-1} binomial(n,k)*a(k). - Robert Israel, Aug 11 2015
a(n) ~ 4*n! / (log(2))^(n+1). - Vaclav Kotesovec, Sep 27 2017
a(n) = Sum_{k>=0} (k + 3)^n / 2^(k+1). - Ilya Gutkovskiy, Jun 27 2020
a(n) = 8*A000670(n) - (2^n + 2 + 4*0^n). - Seiichi Manyama, Dec 21 2023
MAPLE
S:= series(exp(3*x)/(2-exp(x)), x, 31):
seq(coeff(S, x, j)*j!, j=0..30); # Robert Israel, Aug 11 2015
MATHEMATICA
Range[0, 25]! CoefficientList[Series[E^(3 x)/(2 - E^(x)), {x, 0, 25}], x] (* Vincenzo Librandi, Jul 06 2015 *)
PROG
(PARI) { my(x = xx + O(xx^40)); Vec(serlaplace(exp(3*x)/(2-exp(x)))) } \\ Michel Marcus, Jul 06 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sithembele Nkonkobe, Jul 02 2015
EXTENSIONS
More terms from Michel Marcus, Jul 06 2015
STATUS
approved