A092255 -id:A092255 - cp's OEIS frontend

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.

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

Offset: 0

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

nonn

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

			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

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

Cf. A092255.

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

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

A171999 Triangle of trinomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 1, 5, 10, 20, 30, 1, 6, 15, 20, 30, 60, 90, 1, 7, 21, 35, 42, 105, 140, 210, 1, 8, 28, 56, 70, 168, 280, 420, 560, 1, 9, 36, 72, 84, 126, 252, 504, 630, 756, 1260, 1680, 1, 10, 45, 90, 120, 210, 252, 360, 840, 1260, 2520, 3150, 4200, 1, 11, 55

Offset: 1

Clark Kimberling, Jan 21 2010

Number of numbers in row n is given by A086753.
Row sums: A092255.
See A046816, Pascal's pyramid of trinomial coefficients.

			First six rows:
1
1...2
1...3...6
1...4...6....12
1...5...10...20...30
1...6...15...20...30...60...90

Cf. A036038, A046816, A092255.

cp's OEIS Frontend

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.

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