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A230137 a(n)/2^n is the expected value of the maximum of the number of heads and the number of tails when n fair coins are tossed. 2
0, 2, 6, 18, 44, 110, 252, 588, 1304, 2934, 6380, 14036, 30120, 65260, 138712, 297240, 627248, 1332902, 2796876, 5904516, 12333320, 25899972, 53897096, 112693928, 233776464, 487034300, 1007623032, 2092755528, 4319728784, 8948009624, 18432890160, 38094639664 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Georg Braun, On the Growth of a Ballistic Deposition Model on Finite Graphs, arXiv:2001.09836 [math.PR], 2020.

D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.

FORMULA

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

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

a(n) = 2*n/(n-1)*a(n-1) +4*(n-3)/(n-2)*a(n-2) -8*a(n-3) for n>2, else a(n) = n*(1+n). - Alois P. Heinz, Oct 10 2013

From Vaclav Kotesovec, Jul 20 2019: (Start)

a(2*n) = (4^n + binomial(2*n,n))*n.

a(2*n+1) = (4^n + binomial(2*n,n))*(2*n+1). (End)

EXAMPLE

a(2) = 6 because there are four possible events when 2 coins are tossed: HH, HT, TH, TT.  The maximum of the number of heads and number of tails is respectively: 2 + 1 + 1 + 2 = 6.

MAPLE

a:= proc(n) option remember; `if`(n<3, n*(1+n),

       2*n/(n-1)*a(n-1) +4*(n-3)/(n-2)*a(n-2) -8*a(n-3))

    end:

seq(a(n), n=0..40);  # Alois P. Heinz, Oct 10 2013

MATHEMATICA

nn=15; even=Table[n 2^(2n)+n Binomial[2n, n], {n, 0, nn}]; odd=Table[2Sum[ Binomial[2n+1, k]k, {k, n+1, 2n+1}], {n, 0, nn}]; Riffle[even, odd]

CROSSREFS

Sequence in context: A233531 A320303 A319415 * A120414 A251685 A341490

Adjacent sequences:  A230134 A230135 A230136 * A230138 A230139 A230140

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Oct 10 2013

STATUS

approved

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Last modified September 23 09:53 EDT 2021. Contains 347612 sequences. (Running on oeis4.)