A221209 -id:A221209 - cp's OEIS frontend

A225125 Decimal expansion of Integral_{x=0..Pi/2} x^3*cosec(x) dx.

1, 6, 9, 2, 9, 9, 2, 4, 6, 8, 4, 1, 3, 6, 0, 1, 2, 4, 4, 6, 7, 8, 0, 1, 3, 8, 3, 4, 8, 9, 8, 1, 0, 8, 7, 0, 8, 0, 7, 8, 6, 9, 8, 6, 7, 1, 5, 6, 8, 0, 7, 2, 3, 4, 9, 5, 6, 8, 8, 0, 1, 5, 7, 7, 8, 9, 4, 7, 6, 4, 3, 7, 2, 1, 3, 1, 9, 8, 7, 9, 8, 7, 2, 7, 9, 1, 8, 7, 3, 6, 3, 9, 6, 3, 5, 4, 4, 9, 4, 2

Offset: 1

Jean-François Alcover, Apr 30 2013

The simpler Integral_{x=0..Pi/2} x*cosec(x) dx evaluates as 2*Catalan.

			1.6929924684136012446780138348981087080786986715680723495688...

G. C. Greubel, Table of n, a(n) for n = 1..10000
StackExchange, An integral with PolyGamma.

Cf. A006752, A221209.

3*Catalan*Pi^2/2-1/128*(PolyGamma[3, 1/4]-PolyGamma[3, 3/4]); (* or *)
3*Catalan*Pi^2/2-3/64*(Zeta[4, 1/4]-Zeta[4, 3/4]) // RealDigits[#, 10, 100] & // First
RealDigits[Integrate[x^3 Csc[x],{x,0,Pi/2}],10,120][[1]] (* Harvey P. Dale, Jun 16 2025 *)

3*Catalan*Pi^2/2-3/64*(zetahurwitz(4,1/4)-zetahurwitz(4,3/4)) \\ Charles R Greathouse IV, Jan 31 2018

Equals 3*Catalan*Pi^2/2-1/128*(polygamma(3, 1/4)-polygamma(3, 3/4)).

A378910 Decimal expansion of 2*G/Pi, where G = A006752.

Original entry on oeis.org

5, 8, 3, 1, 2, 1, 8, 0, 8, 0, 6, 1, 6, 3, 7, 5, 6, 0, 2, 7, 6, 7, 6, 8, 9, 1, 2, 9, 3, 6, 7, 8, 9, 8, 3, 7, 7, 2, 8, 1, 3, 2, 3, 0, 7, 9, 7, 1, 6, 7, 4, 5, 4, 0, 5, 2, 2, 0, 0, 3, 1, 3, 8, 2, 2, 3, 4, 9, 5, 2, 7, 3, 7, 6, 0, 8, 7, 7, 7, 2, 3, 4, 5, 3, 2, 5, 3, 6, 4, 8, 6, 0, 6, 2, 6, 8, 1, 1, 7, 8

Offset: 0

Stefano Spezia, Dec 10 2024

			0.58312180806163756027676891293678983772813230797167...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.10, p. 232.

Cf. A000796, A006752, A060294, A130834, A143233, A221209, A378911.

Mathematica
```
RealDigits[2Catalan/Pi,10,100][[1]]
```

Equals log(A130834).

Equals 2*A143233.

A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.

Original entry on oeis.org

2, 5, 3, 3, 7, 3, 7, 2, 7, 9, 4, 8, 5, 8, 4, 1, 9, 0, 9, 5, 8, 3, 2, 8, 9, 6, 3, 4, 0, 4, 1, 8, 6, 3, 2, 9, 1, 6, 8, 9, 6, 3, 0, 8, 0, 8, 8, 4, 2, 0, 3, 0, 3, 1, 2, 6, 1, 1, 9, 8, 2, 3, 9, 4, 7, 4, 2, 4, 7, 1, 1, 5, 9, 1, 0, 0, 4, 2, 4, 9, 7, 3, 3, 7, 7, 1, 8, 3, 0, 1, 2, 7, 6, 4, 8, 1, 3, 5, 6, 5

Offset: 1

Stefano Spezia, Dec 10 2024

			2.5337372794858419095832896340418632916896308088420...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.10, p. 233.

Cf. A000796, A002193, A006752, A060294, A130834, A143233, A221209, A378910.

RealDigits[Sqrt[2]*Exp[2Catalan/Pi],10,100][[1]]

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A225125 Decimal expansion of Integral_{x=0..Pi/2} x^3*cosec(x) dx.

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A378910 Decimal expansion of 2*G/Pi, where G = A006752.

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A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.

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cp's OEIS Frontend

A225125 Decimal expansion of Integral_{x=0..Pi/2} x^3*cosec(x) dx.

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A378910 Decimal expansion of 2*G/Pi, where G = A006752.

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A378911 Decimal expansion of sqrt(2)*exp(2*G/Pi), where G = A006752.

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A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.