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A101052 Number of preferential arrangements of n labeled elements when only k<=3 ranks are allowed. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 euro for the taxi, Bob paid 20 euro for the dinner; if they have to equally divide the expenses Alice will have to give 5 euro 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

S. Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938, 2013

Index entries for linear recurrences with constant coefficients, signature (6, -11, 6).

FORMULA

egf = 2*exp(z)-2*exp(z)^2+exp(z)^3; ogf = -(-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

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 *)

CROSSREFS

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

Cf. A000918.

Sequence in context: A008827 A026529 A286182 * A016064 A163774 A304629

Adjacent sequences:  A101049 A101050 A101051 * A101053 A101054 A101055

KEYWORD

nonn

AUTHOR

Thomas Wieder, Nov 28 2004

STATUS

approved

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Last modified December 15 04:23 EST 2019. Contains 329991 sequences. (Running on oeis4.)