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%I #20 May 26 2020 07:53:30
%S 1,6,21,61,158,386,902,2051,4565,10006,21668,46484,98958,209360,
%T 440627,923299,1927456,4010730,8322242,17226050,35578192,73339778,
%U 150918130,310073773,636173403,1303554560,2667935114,5454522188,11140674850,22733861902,46352349432,94435176992
%N Total length squared of longest runs of 1's in all bitstrings of length n.
%C a(n) divided by 2^n is the expected value of the longest run, squared, of heads in n tosses of a fair coin.
%H Steven Finch, <a href="https://arxiv.org/abs/2005.12185">Variance of longest run duration in a random bitstring</a>, arXiv:2005.12185 [math.CO], 2020.
%F O.g.f.: Sum_{k>=1} (2*k-1)*(1/(1-2*x) - (1-x^k)/(1-2*x+x^(k+1))).
%e 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.
%t 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]
%Y Cf. A119706.
%K nonn
%O 1,2
%A _Steven Finch_, May 15 2020
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