A086315 Decimal expansion of constant theta appearing in the expected number of pair of twin vacancies in a digital tree.

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

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.13 Binary search tree constants, p. 356.

Links

Table of n, a(n) for n=1..102.

Eric Weisstein's World of Mathematics, Tree Searching

Formula

Sum_{k>=1} k*2^(k*(k-1)/2))*(Sum_{j=1..k} 1/(2^j-1))/Product_{j=1..k} (2^j-1).

Example

7.743131985536896591446283856749784...

Maple

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

Mathematica

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 *)

Crossrefs

Sequence in context: A198544 A353127 A247274 * A185577 A201517 A010513

Adjacent sequences: A086312 A086313 A086314 * A086316 A086317 A086318

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 15 2003

Status

approved