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A272182
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Decimal expansion of p_3 (so named by S. Finch), a probability related to Vallée's constant.
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0
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1, 0, 2, 7, 8, 1, 6, 4, 7, 7, 9, 0, 6, 6, 5, 9, 6, 4, 3, 2, 3, 8, 2, 9, 5, 3, 4, 0, 9, 8, 7, 8, 3, 2, 5, 4, 4, 6, 3, 5, 4, 7, 1, 5, 8, 4, 2, 3, 7, 6, 3, 8, 9, 0, 9, 2, 3, 7, 3, 2, 4, 0, 5, 4, 3, 9, 6, 1, 9, 1, 6, 2, 9, 1, 3, 2, 2, 9, 2, 8, 9, 9, 9, 4, 0, 9, 5
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OFFSET
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-1,3
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.19 Vallée's Constant, p. 161.
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LINKS
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FORMULA
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Sum_{i >= 1, j >= 1, k >= 1} 1 / ((i*j*k + i + k)^2 * (i*j*k + i*j + i + k + 1)^2).
Equals Sum_{i >= 1, j >= 1} (2*PolyGamma(1,i/(i*j + 1) + 1) - ((i*j + 1) * (i*(3*j + 2) + 3))/(i*j + i + 1)^2)/(i*j + 1)^4, where PolyGamma(1,x) is
the derivative of the digamma function psi(x).
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EXAMPLE
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0.010278164779066596432382953409878325446354715842376389092373240543961916291...
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MATHEMATICA
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p3 = NSum[(2*PolyGamma[1, i/(i*j + 1) + 1] - ((i*j + 1)*(i*(3*j + 2) + 3))/(i*j + i + 1)^2)/(i*j + 1)^4, {i, 1, 10^4}, {j, 1, 10^4 - i}, NSumTerms -> 10^3]; Join[{0}, RealDigits[p3, 10, 10][[1]]]
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PROG
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(Magma) nMax:=120; M:=75; SetDefaultRealField(RealField(160)); S:=[]; sum:=0.0; for n in [1..nMax] do for i in [1..n] do for j in [1..n] do if (i eq n) or (j eq n) then kMin:=1; else kMin:=n; end if; for k in [kMin..n] do sum:=sum+1/((i*j*k + i + k)^2 * (i*j*k + i*j + i + k + 1)^2); end for; end for; end for; S[n]:=sum; end for; t:=[]; for n in [1..#S] do t[n]:=S[n]*n^M; end for; for m in [1..M] do for j in [1..#S-m] do t[j]:=(t[j+1]-t[j])/m; if m eq M then ChangePrecision(t[j], 90); end if; end for; end for; // Jon E. Schoenfield, Feb 18 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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