cp's OEIS Frontend

A101052 Number of preferential arrangements of n labeled elements when only k <= 3 ranks are allowed.

%I A101052 #38 Dec 26 2024 12:29:01
%S A101052 1,1,3,13,51,181,603,1933,6051,18661,57003,173053,523251,1577941,
%T A101052 4750203,14283373,42915651,128878021,386896203,1161212893,3484687251,
%U A101052 10456158901,31372671003,94126401613,282395982051,847221500581
%N A101052 Number of preferential arrangements of n labeled elements when only k <= 3 ranks are allowed.
%C A101052 The (labeled) case for k <= 2 is given by A000225. The unlabeled analog for k <= 2 is given by A028310 (A000027). The unlabeled analog for k <= 3 is given by A000124.
%C A101052 Alice and Bob went out for dinner; Alice paid 10 euros for the taxi, Bob paid 20 euros for the dinner; if they have to equally divide the expenses Alice will have to give 5 euros to Bob. With two people, Alice and Bob, there are three possible cases: Alice has to give money to Bob; Bob has to give money to Alice; they paid the same amount, so no debtors nor creditors. With three people, there are 13 cases; with four people, there are 51 cases, and so on. - Alessandro Gentilini (alessandro.gentilini(AT)gmail.com), Aug 10 2006
%H A101052 Vincenzo Librandi, <a href="/A101052/b101052.txt">Table of n, a(n) for n = 0..1000</a>
%H A101052 Hongyi Cao, Gang Xu, Renshu Gu, Jinlan Xu, Xiaoyu Zhang, Timon Rabczuk, Yuzhe Luo, and Xifeng Gao, <a href="https://arxiv.org/abs/2412.15564">Robust and Feature-Preserving Offset Meshing</a>, arXiv:2412.15564 [cs.GR], 2024. See p. 14.
%H A101052 Samuele Giraudo, <a href="http://arxiv.org/abs/1306.6938">Combinatorial operads from monoids</a>, arXiv preprint arXiv:1306.6938, 2013
%H A101052 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F A101052 E.g.f. = 2*exp(z) - 2*exp(z)^2 + exp(z)^3;
%F A101052 o.g.f. = -(-1+3*z-6*z^2)/(11*z^2+1-6*z-6*z^3).
%F A101052 a(n) = 3^n + 2 - 2*2^n; recurrence: a(n+3) - 6*a(n+2) + 11*a(n+1) - 6*a(n), a(0) = 1, a(1) = 1, a(2) = 3.
%F A101052 G.f.: Sum_{n>=0} a(n)*log(1+x)^n/n! = (1-x^4)/(1-x). - _Paul D. Hanna_, Feb 18 2012
%F A101052 Binomial transform of A000918 in which the first term is changed from -1 to 1 as: (1, 0, 2, 6, 14, 30, 62, ...). - _Gary W. Adamson_, Mar 23 2012
%p A101052 A101052 := n -> 3^n+2-2*2^n; [ seq(3^n+2-2*2^n,n=0..30) ];
%t A101052 a = Exp[x] - 1;CoefficientList[Series[1+a+a^2+a^3,{x,0,20}],x]*Table[n!,{n,0,20}]
%t A101052 LinearRecurrence[{6,-11,6},{1,1,3},30] (* _Harvey P. Dale_, Mar 13 2013 *)
%Y A101052 Cf. A000670, A000225, A000124, A028310, A097237.
%Y A101052 Cf. A000918.
%K A101052 nonn
%O A101052 0,3
%A A101052 _Thomas Wieder_, Nov 28 2004