OFFSET
1,2
COMMENTS
a(n) divided by 2^n is the expected value of the longest run, squared, of heads in n tosses of a fair coin.
LINKS
Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.
FORMULA
O.g.f.: Sum_{k>=1} (2*k-1)*(1/(1-2*x) - (1-x^k)/(1-2*x+x^(k+1))).
EXAMPLE
a(3)=21 because for the 8(2^3) possible runs 0 is longest run of heads once, 1 four times, 2 two times and 3 once and 0*1+1*4+4*2+9*1 = 21.
MATHEMATICA
nn = 10; Drop[Apply[Plus, Table[CoefficientList[Series[(2 n - 1) (1/(1 - 2 x) - (1 - x^n)/(1 - 2 x + x^(n + 1))), {x, 0, nn}], x], {n, 1, nn}]], 1]
CROSSREFS
KEYWORD
nonn
AUTHOR
Steven Finch, May 15 2020
STATUS
approved