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A086315
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Decimal expansion of constant theta appearing in the expected number of pair of twin vacancies in a digital tree.
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1
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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, 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, 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
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.13 Binary search tree constants, p. 356.
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LINKS
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FORMULA
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Sum_{k>=1} k*2^(k*(k-1)/2))*(Sum_{j=1..k} 1/(2^j-1))/Product_{j=1..k} (2^j-1).
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EXAMPLE
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7.743131985536896591446283856749784...
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MAPLE
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theta:= sum((((k*2^(k*(k-1)/2)) *sum(1/(2^j-1), j=1..k))/
product(2^j-1, j=1..k)), k=1..infinity):
s:= convert(evalf(theta, 110), string):
map(parse, subs("."=[][], [seq(i, i=s)]))[]; # Alois P. Heinz, Jun 27 2014
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MATHEMATICA
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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 *)
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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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