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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 (list; graph; refs; listen; history; text; internal format)
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 equally-likely 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)^3-3*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!(n-i-j)!), j=i+1..floor((n-i)/2)), j=0..floor((n-2)/3)). - Dennis P. Walsh, Jun 19 2013

EXAMPLE

For n=4, a(4)=15 since there are 15 five-letter 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+1-i, j), j=i+1..floor((n-i)/2)), i=0..floor((n-2)/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

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Last modified June 3 05:29 EDT 2020. Contains 334798 sequences. (Running on oeis4.)