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A272182 Decimal expansion of p_3 (so named by S. Finch), a probability related to Vallée's constant. 0
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)