

A115752


Number of words of length n+1 created with the letters a,b,c with more c's than b's and more b's than a's.


1



0, 0, 3, 4, 15, 81, 168, 540, 2271, 5365, 16698, 63229, 159250, 489048, 1749933, 4576140, 13955895, 48211389, 129211818, 392441049, 1323741156, 3609608838, 10933915743, 36252591813, 100126350090, 302737691646, 990855646563
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OFFSET

0,3


COMMENTS

Also, if n+1 voters vote for one of the three candidates (A, B, or C) in an election, a(n) is the number of possible ballot results in which candidate C gets more votes than candidate B and candidate B gets more votes than candidate A. We note that the number of all possible ballot results is 3^(n+1). Hence, if all three candidates are equallylikely to get a random voter's vote, the probability of no ties among any of the candidates is 3!*a(n)/3^(n+1).  Dennis P. Walsh, Jun 19 2013


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Mike Zabrocki, Math 5020, York University


FORMULA

E.g.f.: (t(1)^33*t(1)*t(2)+2*t(3))/6 where t(1)=hypergeom([],[],x), t(2)=hypergeom([],[1],x^2) and t(3)=hypergeom([],[1,1],x^3).  Vladeta Jovovic, Sep 22 2007
a(n) = sum(sum(n!/(i!j!(nij)!), j=i+1..floor((ni)/2)), j=0..floor((n2)/3)).  Dennis P. Walsh, Jun 19 2013


EXAMPLE

For n=4, a(4)=15 since there are 15 fiveletter words with more c's than b's and more b's than a's. Ten of the words use 3 c's and 2 b's, namely, cccbb, ccbcb, ccbbc, cbccb, cbcbc, cbbcc, bcccb, bccbc, bcbcc, and bbccc; and 5 of the words use 4 c's and 1 b, namely, ccccb, cccbc, ccbcc, cbccc, and bcccc.  Dennis P. Walsh, Jun 19 2013


MAPLE

seq(add(binomial(n+1, i)*add(binomial(n+1i, j), j=i+1..floor((ni)/2)), i=0..floor((n2)/3)), n=0..30); # Dennis P. Walsh, Jun 19 2013


CROSSREFS

Cf. A092255.
Sequence in context: A171062 A171061 A331869 * A103095 A041325 A042541
Adjacent sequences: A115749 A115750 A115751 * A115753 A115754 A115755


KEYWORD

nonn


AUTHOR

Kevin Smith (kjsmith(AT)yorku.ca), Mar 28 2006


STATUS

approved



