A242191 Expected value of the highest die when n six-sided dice are rolled, multiplied by 6^n.

21, 161, 1071, 6797, 42231, 259421, 1582791, 9614717, 58230711, 351922781, 2123580711, 12799240637, 77074749591, 463808234141, 2789504205831, 16769733474557, 100779708074871, 605475935585501, 3636808913042151, 21840480209276477, 131140458175102551, 787328413691288861

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..1282

Formula

a(n) = 1 + Sum_{k=2..6, j=1..n} k*binomial(n,j)*(k-1)^(n-j).

Conjecture: a(n) = -5^n-3^n+6^(n+1)-2^n-1-4^n with generating function -x*(-21 +280*x -1365*x^2 +2954*x^3 -2688*x^4 +720*x^5) / ( (x-1) *(6*x-1) *(4*x-1) *(3*x-1) *(2*x-1) *(5*x-1) ) - R. J. Mathar, May 23 2014

Conjecture is true, because the probability that the highest is k is (k/6)^n - ((k-1)/6)^n for k = 1..6. - Robert Israel, Mar 09 2020

Example

a(1) = 21, because when a die is rolled, the possible outcomes are 1,2,3,4,5,6, whose average is 21/6.
a(2) = 161 because when two dice are rolled, the expected value of the higher die is 161/36.

Maple

f:= n -> 6^(n+1)-add(i^n, i=1..5):
map(f, [$1..50]); # Robert Israel, Mar 09 2020

Crossrefs

Sequence in context: A267473 A179097 A070315 * A146301 A338891 A126993

Adjacent sequences: A242188 A242189 A242190 * A242192 A242193 A242194

Keyword

nonn

Author

Andrew Woods, May 06 2014

Status

approved