Iaroslav V. Blagouchine - cp's OEIS frontend

A302121 Denominators of a series converging to Euler's constant.

4, 96, 72, 46080, 1152, 2322432, 100352, 7431782400, 2090188800, 2452488192000, 2697737011200, 64274810535936000, 2923954176000, 1799694695006208000, 3085190905724928, 33566877054287216640000, 4458100858772520960000, 120538655501945394954240000, 1057781497894797312000

Offset: 1

Iaroslav V. Blagouchine, Apr 01 2018

gamma = 3/4 - 11/96 - 1/72 - 311/46080 - 5/1152 - 7291/2322432 - ..., see formula (104) in the reference below.

			Denominators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...

G. C. Greubel, Table of n, a(n) for n = 1..250
Ia. V. Blagouchine, Three Notes on Ser's and Hasse's Representations for the Zeta-functions. INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A3, pp. 1-45, 2018. arXiv:1606.02044 [math.NT], 2016.

Cf. A302120 (numerators of this series), A262856, A262858.

a := proc (n) options operator, arrow; denum((1/2)*(-1)^(n+1)*(sum(Stirling1(n-1, l)*((-1/2)^(l+1)+1)/(l+1), l = 0 .. n-1))/factorial(n)+(-1)^(n+1)*(sum(Stirling1(n, l)/(l+1), l = 1 .. n))/(n*factorial(n))) end proc

a[n_] := Denominator[(1/2)*(-1)^(n+1)*(Sum[StirlingS1[n-1,l]*((-1/2)^(l+1) + 1)/(l+1),{l,0,n-1}])/(n!) + (-1)^(n+1)*(Sum[StirlingS1[n, l]/(l+1),{l,1,n}])/(n*n!)]; Table[a[n], {n, 1, 24}]

a(n) = denominator((1/2)*(-1)^(n+1)*(sum(l=0,n-1,stirling(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(sum(l=1,n,stirling(n,l)/(l+1)))/(n*n!))

a(n) = Denominators of ((1/2)*(-1)^(n+1)*(Sum_{l=0..n-1} (S_1(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(Sum_{l=1..n} S_1(n,l)/(l+1)))/(n*n!)), where S_1(x,y) are the signed Stirling numbers of the first kind.

A302120 Absolute value of the numerators of a series converging to Euler's constant.

Original entry on oeis.org

3, 11, 1, 311, 5, 7291, 243, 14462317, 3364621, 3337014731, 3155743303, 65528247068741, 2627553901, 1439156737843967, 2213381206625, 21757704362231905789, 2627003970197650333, 64925181492079668050329, 523317843775891637, 161371847993975070290712761, 78461950306245817433389909

Offset: 1

Iaroslav V. Blagouchine, Apr 01 2018

gamma = 3/4 - 11/96 - 1/72 - 311/46080 - 5/1152 - 7291/2322432 - ..., see formula (104) in the reference below.

			Numerators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...

G. C. Greubel, Table of n, a(n) for n = 1..250
Ia. V. Blagouchine, Three Notes on Ser's and Hasse's Representations for the Zeta-functions. INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A3, pp. 1-45, 2018. arXiv:1606.02044 [math.NT], 2016.

Cf. A302121 (denominators of this series), A262856, A262858.

[3] cat [Abs(Numerator( (1/2)*(-1)^(n+1)*(&+[StirlingFirst(n-1,k)*((-1/2)^(k+1) + 1)/(k+1): k in [1..n-1]])/Factorial(n) + (-1)^(n+1)*(&+[StirlingFirst(n,k)/(k+1): k in [1..n]])/(n*Factorial(n)) )): n in [2..30]]; // G. C. Greubel, Oct 29 2018

a:= proc(n) abs(numer((1/2)*(-1)^(n+1)*(add(Stirling1(n-1, l)*((-1/2)^(l+1)+1)/(l+1), l = 0 .. n-1))/(n)!+(-1)^(n+1)*(add(Stirling1(n, l)/(l+1), l = 1 .. n))/(n*(n)!))) end proc: seq(a(n), n=1..23);

a[n_] := Numerator[(1/2)*(-1)^(n+1)*(Sum[StirlingS1[n-1,l]*((-1/2)^(l+1) + 1)/(l+1),{l,0,n-1}])/(n!) + (-1)^(n+1)*(Sum[StirlingS1[n, l]/(l+1),{l,1,n}])/(n*n!)]; Table[Abs[a[n]], {n, 1, 24}]

a(n) = abs(numerator((1/2)*(-1)^(n+1)*(sum(l=0,n-1,stirling(n-1,l)*((-1/2)^(l+1) + 1)/(l+1))) /(n!) + (-1)^(n+1)*(sum(l=1,n,stirling(n,l)/(l+1)))/(n*n!)))

a(n) = abs(Numerators of ((1/2)*(-1)^(n+1)*(Sum_{l=0,n-1} (S_1(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(Sum_{l=1,n} S_1(n,l)/(l+1)))/(n*n!))), where S_1(x,y) are the signed Stirling numbers of the first kind.

A300731 Decimal expansion of sqrt(Pi^4/96 - 1).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

Also the total harmonic distortion (THD) of a triangle wave, see formula (14) in the Blagouchine & Moreau link.

			0.1211529265193047433149738747453528509885975440568532...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues, IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.
Index entries for transcendental numbers

Cf. A092425, A300690, A300707, A300713, A300714, A300727.

MATLAB
```
format long; sqrt(pi^4/96-1)
```
Maple
```
evalf(sqrt((1/96)*Pi^4-1), 120)
```

RealDigits[Sqrt[Pi^4/96 - 1], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^4/96-1)

A300714 Decimal expansion of the total harmonic distortion (THD) of the sawtooth signal filtered by a 1st-order low-pass filter.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

See formula (33) in the Blagouchine & Moreau link.

			0.3694861820952043731200546914239769936602361586790838...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.

Cf. A195055, A300690, A300713, A300727.

MATLAB
```
format long; sqrt(pi^2/3-pi*coth(pi))
```

evalf(sqrt((1/3)*Pi^2-Pi*coth(Pi)), 120)

RealDigits[Sqrt[Pi^2/3 - Pi*Coth[Pi]], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^2/3-Pi/tanh(Pi))

Equals sqrt(Pi^2/3 - Pi*coth(Pi)).

A300713 Decimal expansion of sqrt(Pi^2/6 - 1) = sqrt(zeta(2) - 1).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

Also the total harmonic distortion (THD) of a sawtooth signal, see formula (15) in the Blagouchine & Moreau link.

			0.8030778709740584281843212446690348231898910996409661...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.
Index entries for transcendental numbers

Cf. A013661, A300690, A300714, A300727, A300731.

MATLAB
```
format long; sqrt(pi^2/6-1)
```
Maple
```
evalf(sqrt((1/6)*Pi^2-1), 120)
```

RealDigits[Sqrt[Pi^2/6 - 1], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^2/6-1)

Equals sqrt(A013661 - 1).

A300710 Decimal expansion of 17*Pi^8/161280.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^8), whose value is obtained from zeta(8) given by L. Euler in 1735: Sum_{n>=0} (2n+1)^(-s)=(1-2^(-s))*zeta(s).

			1.0001551790252961193029872492957280415665429750613740...

Jason Bard, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A092736, A111003, A300707, A300709.

MATLAB
```
format long; (17/161280)*pi^8
```
Maple
```
evalf((17/161280)*Pi^8, 120);
```

RealDigits[(17/161280)*Pi^8, 10, 120][[1]]

default(realprecision, 120); (17/161280)*Pi^8

Equals 17*A092736/161280. - Omar E. Pol, Mar 11 2018
From Artur Jasinski, Jun 24 2025: (Start)
Equals DirichletL(2,1,8).
Equals DirichletL(4,1,8).
Equals DirichletL(8,1,8).
Equals DirichletL(16,1,8). (End)
Equals 255*Zeta(8)/256. - Jason Bard, Aug 21 2025

A300709 Decimal expansion of Pi^6/960.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^6), whose value is obtained from zeta(6) given by L. Euler in 1735: Sum_{n>=0}(2n+1)^(-s) = (1-2^(-s))*zeta(s).

			1.0014470766409421219064785871379373946533515917510902...

Index entries for transcendental numbers

Cf. A092732, A111003, A300707, A300710.

MATLAB
```
format long; pi^6/960
```
Maple
```
evalf((1/960)*Pi^6, 120)
```
Mathematica
```
RealDigits[Pi^6/960, 10, 120][[1]]
```
PARI
```
default(realprecision, 120); Pi^6/960
```

Equals A092732/960. - Omar E. Pol, Mar 11 2018
From Artur Jasinski, Jun 24 2025: (Start)
Equals DirichletL(2,1,6).
Equals DirichletL(4,1,6).
Equals DirichletL(8,1,6).
Equals DirichletL(16,1,6). (End)

A300707 Decimal expansion of Pi^4/96.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^4), whose value is obtained from zeta(4) given by L. Euler in 1735: Sum_{n>=0} (2n+1)^(-s) = (1-2^(-s))*zeta(s).

For the partial sums of this series see A120269/A128493. - Wolfdieter Lang, Sep 02 2019

			1.0146780316041920545462534655073449088513290174238064...

Eric Weisstein's World of Mathematics, Dirichlet Lambda Function. See (6).
Index entries for transcendental numbers

Cf. A013662, A092425, A111003, A120269, A128493, A300709, A300710, A300731.

MATLAB
```
format long; pi^4/96
```
Maple
```
evalf((1/96)*Pi^4, 120)
```
Mathematica
```
RealDigits[Pi^4/96, 10, 120][[1]]
```
PARI
```
default(realprecision, 120); Pi^4/96
```

Equals A092425/96. - Omar E. Pol, Mar 11 2018
Equals (15/16)*zeta(4) = (15/16)*A013662. - Wolfdieter Lang, Sep 02 2019
Equals Sum_{k>=1} 1/(2*k-1)^4. - Sean A. Irvine, Mar 25 2025
Equals lambda(4), where lambda is the Dirichlet lambda function. - Michel Marcus, Aug 15 2025

A300690 Decimal expansion of sqrt(Pi^2/8 - 1).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

Also the total harmonic distortion (THD) of a square wave, see formula (11) in the Blagouchine & Moreau link.

			0.4834258476086790990137326370639317022328017276651459...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.
Index entries for transcendental numbers

Cf. A002388, A111003, A300713, A300714, A300727, A300731.

MATLAB
```
format long; sqrt(pi^2/8-1)
```
Maple
```
evalf(sqrt((1/8)*Pi^2-1), 120)
```

RealDigits[Sqrt[Pi^2/8 - 1], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^2/8-1)

A300727 Decimal expansion of the total harmonic distortion (THD) of the sawtooth signal filtered by a 2nd-order low-pass filter.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

See formula (34) in Blagouchine & Moreau link.

			0.1811416137938290040802181055813016784438928351595635...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.

Cf. A247719 (Pi/sqrt(2)), A300690, A300713, A300714.

format long; sqrt(sqrt(pi*(cot(pi/sqrt(2))*coth(pi/sqrt(2))^2-cot(pi/sqrt(2))^2*coth(pi/sqrt(2))-cot(pi/sqrt(2))-coth(pi/sqrt(2)))/((cot(pi/sqrt(2))^2+coth(pi/sqrt(2))^2)*sqrt(2))+(1/3)*pi^2-1))

evalf(sqrt(Pi*(cot(Pi/sqrt(2))*coth(Pi/sqrt(2))^2-cot(Pi/sqrt(2))^2*coth(Pi/sqrt(2))-cot(Pi/sqrt(2))-coth(Pi/sqrt(2)))/((cot(Pi/sqrt(2))^2+coth(Pi/sqrt(2))^2)*sqrt(2))+(1/3)*Pi^2-1), 120)

RealDigits[Sqrt[Pi*(Cot[Pi/Sqrt[2]]*Coth[Pi/Sqrt[2]]^2-Cot[Pi/Sqrt[2]]^2*Coth[Pi/Sqrt[2]]-Cot[Pi/Sqrt[2]]-Coth[Pi/Sqrt[2]])/((Cot[Pi/Sqrt[2]]^2+Coth[Pi/Sqrt[2]]^2)*Sqrt[2])+(1/3)*Pi^2-1], 10, 120][[1]]

s2=sqrt(2);
A=Pi/s2;
B=1+2/(exp(2*A)-1)
C=1/tan(A);
sqrt(Pi*(B^2*C-B*C^2-C-B)/((C^2+B^2)*s2) + Pi^2/3 - 1) \\ Charles R Greathouse IV, Mar 11 2018

Equals sqrt(Pi*(cot(Pi/sqrt(2))*coth(Pi/sqrt(2))^2-cot(Pi/sqrt(2))^2*coth(Pi/sqrt(2))-cot(Pi/sqrt(2))-coth(Pi/sqrt(2)))/((cot(Pi/sqrt(2))^2+coth(Pi/sqrt(2))^2)*sqrt(2))+(1/3)*Pi^2-1).

cp's OEIS Frontend

User: Iaroslav V. Blagouchine

A302121 Denominators of a series converging to Euler's constant.

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A302120 Absolute value of the numerators of a series converging to Euler's constant.

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A300731 Decimal expansion of sqrt(Pi^4/96 - 1).

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A300714 Decimal expansion of the total harmonic distortion (THD) of the sawtooth signal filtered by a 1st-order low-pass filter.

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A300713 Decimal expansion of sqrt(Pi^2/6 - 1) = sqrt(zeta(2) - 1).

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A300710 Decimal expansion of 17*Pi^8/161280.

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