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A258749
Decimal expansion of Ls_3(Pi), the value of the 3rd basic generalized log-sine integral at Pi (negated).
11
2, 5, 8, 3, 8, 5, 6, 3, 9, 0, 0, 2, 4, 9, 8, 5, 0, 1, 4, 6, 2, 3, 0, 2, 6, 2, 5, 5, 5, 9, 1, 7, 8, 2, 9, 3, 3, 5, 1, 8, 7, 7, 4, 0, 4, 7, 1, 5, 7, 0, 9, 2, 3, 0, 7, 8, 4, 5, 3, 7, 8, 1, 7, 5, 3, 1, 7, 1, 9, 9, 5, 7, 6, 4, 5, 5, 4, 7, 5, 5, 0, 3, 1, 3, 0, 5, 5, 8, 4, 1, 9, 3, 8, 3, 5, 7, 3, 8, 4, 9, 4, 1, 9
OFFSET
1,1
FORMULA
-Integral_{0..Pi} log(2*sin(t/2))^2 dx = -Pi^3/12.
Also equals 2nd derivative of -Pi*binomial(x, x/2) at x=0.
It can be noticed that Ls_2(Pi) is 0, and that Ls_2(Pi/2) is Catalan's constant 0.915966... (A006752).
EXAMPLE
-2.5838563900249850146230262555917829335187740471570923078453781753171...
MATHEMATICA
RealDigits[-Pi^3/12, 10, 103] // First
PROG
(PARI) -Pi^3/12 \\ G. C. Greubel, Aug 23 2018
(Magma) R:= RealField(100); -Pi(R)^3/12; // G. C. Greubel, Aug 23 2018
CROSSREFS
Cf. A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).
Sequence in context: A220398 A352633 A200225 * A056886 A197839 A021391
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved