cp's OEIS Frontend

A259533 Number of restricted barred preferential arrangements of an n-set having 3 bars in which 3 fixed sections are restricted sections and 1 section is a free section.

%I A259533 #60 Nov 15 2024 01:40:44
%S A259533 1,4,18,94,582,4294,37398,378214,4366422,56697574,817979478,
%T A259533 12981058534,224732536662,4214866778854,85130743747158,
%U A259533 1842265527790054,42525237455785302,1042966136232956134,27084277306054500438,742412698554626764774,21421502369955072576342,648998599988032588957414
%N A259533 Number of restricted barred preferential arrangements of an n-set having 3 bars in which 3 fixed sections are restricted sections and 1 section is a free section.
%C A259533 Also, number of preferential fuzzy subsets of length n+2 where the keychains are of length n+2.
%C A259533 Binomial transform of A007047.
%C A259533 Double binomial transform of A000629.
%C A259533 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
%H A259533 Robert Israel, <a href="/A259533/b259533.txt">Table of n, a(n) for n = 0..387</a>
%H A259533 V. Murali, <a href="http://www.pphmj.com/abstract/1470.htm">Ordered partitions and finite fuzzy sets</a>, Far East J. Math. Sci.(FJMS), 21(2006), 12-132.
%H A259533 S. Nkonkobe and V. Murali, <a href="http://arxiv.org/abs/1503.06172">A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements</a>, arXiv:1503.06172 [math.CO], 2015.
%F A259533 E.g.f.: exp(3*x)/(2-exp(x)).
%F A259533 a(n) = 3^n + Sum_{k = 0..n-1} binomial(n,k)*a(k). - _Robert Israel_, Aug 11 2015
%F A259533 a(n) ~ 4*n! / (log(2))^(n+1). - _Vaclav Kotesovec_, Sep 27 2017
%F A259533 a(n) = Sum_{k>=0} (k + 3)^n / 2^(k+1). - _Ilya Gutkovskiy_, Jun 27 2020
%F A259533 a(n) = 8*A000670(n) - (2^n + 2 + 4*0^n). - _Seiichi Manyama_, Dec 21 2023
%p A259533 S:= series(exp(3*x)/(2-exp(x)),x,31):
%p A259533 seq(coeff(S,x,j)*j!, j=0..30); # _Robert Israel_, Aug 11 2015
%t A259533 Range[0, 25]! CoefficientList[Series[E^(3 x)/(2 - E^(x)), {x, 0, 25}], x] (* _Vincenzo Librandi_, Jul 06 2015 *)
%o A259533 (PARI) { my(x = xx + O(xx^40)); Vec(serlaplace(exp(3*x)/(2-exp(x)))) } \\ _Michel Marcus_, Jul 06 2015
%Y A259533 Cf. A000670, A000629, A007047.
%K A259533 nonn,easy
%O A259533 0,2
%A A259533 _Sithembele Nkonkobe_, Jul 02 2015
%E A259533 More terms from _Michel Marcus_, Jul 06 2015