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

1, 1, 3, 13, 51, 181, 603, 1933, 6051, 18661, 57003, 173053, 523251, 1577941, 4750203, 14283373, 42915651, 128878021, 386896203, 1161212893, 3484687251, 10456158901, 31372671003, 94126401613, 282395982051, 847221500581

Offset: 0

Author

Thomas Wieder, Nov 28 2004

Keywords

• nonn

Comments

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.

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Hongyi Cao, Gang Xu, Renshu Gu, Jinlan Xu, Xiaoyu Zhang, Timon Rabczuk, Yuzhe Luo, and Xifeng Gao, Robust and Feature-Preserving Offset Meshing, arXiv:2412.15564 [cs.GR], 2024. See p. 14.
• Samuele Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938, 2013
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A000670, A000225, A000124, A028310, A097237.

Cf. A000918.

Programs

Maple

A101052 := n -> 3^n+2-2*2^n; [ seq(3^n+2-2*2^n,n=0..30) ];

Mathematica

a = Exp[x] - 1;CoefficientList[Series[1+a+a^2+a^3,{x,0,20}],x]*Table[n!,{n,0,20}]
LinearRecurrence[{6,-11,6},{1,1,3},30] (* Harvey P. Dale, Mar 13 2013 *)

Formula

E.g.f. = 2*exp(z) - 2*exp(z)^2 + exp(z)^3;

o.g.f. = -(-1+3*z-6*z^2)/(11*z^2+1-6*z-6*z^3).

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.

G.f.: Sum_{n>=0} a(n)*log(1+x)^n/n! = (1-x^4)/(1-x). - Paul D. Hanna, Feb 18 2012

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