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%I #15 Nov 19 2015 10:29:10
%S 7,7,4,3,1,3,1,9,8,5,5,3,6,8,9,6,5,9,1,4,4,6,2,8,3,8,5,6,7,4,9,7,8,4,
%T 2,9,5,5,9,3,6,5,2,8,2,8,4,1,8,8,0,8,8,8,8,6,6,5,1,8,5,5,9,1,8,3,8,3,
%U 2,9,9,7,1,5,1,7,6,2,9,2,9,0,1,5,1,0,9,4,3,9,0,7,9,9,5,5,4,3,5,6,7,5
%N Decimal expansion of constant theta appearing in the expected number of pair of twin vacancies in a digital tree.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.13 Binary search tree constants, p. 356.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TreeSearching.html">Tree Searching</a>
%F Sum_{k>=1} k*2^(k*(k-1)/2))*(Sum_{j=1..k} 1/(2^j-1))/Product_{j=1..k} (2^j-1).
%e 7.743131985536896591446283856749784...
%p theta:= sum((((k*2^(k*(k-1)/2)) *sum(1/(2^j-1), j=1..k))/
%p product(2^j-1, j=1..k)), k=1..infinity):
%p s:= convert(evalf(theta, 110), string):
%p map(parse, subs("."=[][], [seq(i, i=s)]))[]; # _Alois P. Heinz_, Jun 27 2014
%t digits = 102; m0 = 100; dm = 100; Clear[theta]; theta[m_] := theta[m] = Sum[((k*2^(k*((k-1)/2)))*Sum[1/(2^j-1), {j, 1, k}])/Product[2^j-1, {j, 1, k}], {k, 1, m}] // N[#, digits+10]&; theta[m0]; theta[m = m0 + dm]; While[RealDigits[theta[m], 10, digits+10] != RealDigits[theta[m - dm], 10, digits+10], Print["m = ", m]; m = m + dm]; RealDigits[theta[m], 10, digits] // First (* _Jean-François Alcover_, Jun 27 2014 *)
%t digits = 102; theta = NSum[((k*2^(k*((k-1)/2)))*((QPolyGamma[0, 1+k, 1/2] - QPolyGamma[0, 1, 1/2])/Log[2]))/((-1)^k*QPochhammer[2, 2, k]), {k, 1, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 3*digits]; RealDigits[theta, 10, digits] // First (* _Jean-François Alcover_, Nov 19 2015 *)
%K nonn,cons
%O 1,1
%A _Eric W. Weisstein_, Jul 15 2003
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