A272182 Decimal expansion of p_3 (so named by S. Finch), a probability related to Vallée's constant.

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

Offset

-1,3

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.19 Vallée's Constant, p. 161.

Links

Table of n, a(n) for n=-1..85.

Formula

Equals 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).

Example

0.010278164779066596432382953409878325446354715842376389092373240543961916291...

Mathematica

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]]]

Prog

(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

Crossrefs

Cf. A143302, A143303, A145426 (p_1), A247318 (p_2).

Sequence in context: A306417 A011416 A086658 * A175577 A189039 A198815

Adjacent sequences: A272179 A272180 A272181 * A272183 A272184 A272185

Keyword

nonn,cons

Author

Jean-François Alcover, May 17 2016

Extensions

More digits from Jon E. Schoenfield, Feb 18 2018

Status

approved