A091476 -id:A091476 - cp's OEIS frontend

A181411 a(n) = Sum_{k=0..n} C(n,k)*sigma(n+k) for n>=1.

4, 18, 55, 150, 379, 915, 2146, 4934, 11080, 24833, 54476, 119091, 259432, 556700, 1195135, 2561094, 5428597, 11488866, 24350993, 51296325, 107427025, 225330244, 472762497, 985966379, 2049357779, 4267962522, 8887535983, 18431783744

Offset: 1

Paul D. Hanna, Oct 19 2010

nonn

			L.g.f.: L(x) = 4*x + 18*x^2/2 + 55*x^3/3 + 150*x^4/4 + 379*x^5/5 +...
Exponentiation yields the g.f. of A181410:
exp(L(x)) = 1 + 4*x + 17*x^2 + 65*x^3 + 234*x^4 + 804*x^5 +...
The initial terms begin:
a(1) = 1*1 + 1*3 = 4;
a(2) = 1*3 + 2*4 + 1*7 = 18;
a(3) = 1*4 + 3*7 + 3*6 + 1*12 = 55;
a(4) = 1*7 + 4*6 + 6*12 + 4*8 + 1*15 = 150; ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..3000
Vaclav Kotesovec, Plot of a(n)/(n*2^n) for n = 1..10000

Cf. A181410.

Table[Sum[Binomial[n,k] * DivisorSigma[1,n+k],{k,0,n}],{n,1,30}] (* Vaclav Kotesovec, Oct 05 2020 *)

{a(n)=sum(k=0,n,binomial(m,k)*sigma(n+k))}

Equals the logarithmic derivative of A181410.

Conjecture: a(n) ~ c * n * 2^n, where c = Pi^2/4 = A091476. - Vaclav Kotesovec, Oct 05 2020

A363848 Decimal expansion of the arithmetic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

Original entry on oeis.org

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

Offset: 0

Tian Vlasic, Jun 24 2023

The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.

The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.

			0.933174653498462644...

Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral

Cf. A091476, A363874, A363876.

First[RealDigits[Pi^2/4 * NIntegrate[Sqrt[1-x^2]/EllipticE[x^2]^2, {x,0,1}, WorkingPrecision -> 100]]] (* Stefano Spezia, Jun 24 2023 *)

Equals ((Pi^2)/4) * Integral_{x=0..1} sqrt(1 - x^2)/E(x)^2 dx.

More terms from Stefano Spezia, Jun 24 2023

A363874 Decimal expansion of the harmonic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

Original entry on oeis.org

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

Offset: 0

Tian Vlasic, Jun 25 2023

The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.

The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.

			0.87892065082960412...

Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral

Cf. A091476, A363848, A363876.

First[RealDigits[Pi^2/(4 * NIntegrate[EllipticE[x^2]^2/Sqrt[1 - x^2], {x, 0, 1}, WorkingPrecision -> 100])]]

Pi^2/(4*intnum(x=0,1,(ellE(x)^2)/sqrt(1 - x^2))) \\ Hugo Pfoertner, Jun 25 2023

Equals Pi^2/(4*Integral_{x=0..1} (E(x)^2)/sqrt(1 - x^2) dx).

A073347 a(1)=1; a(n+1) is the smallest integer > a(n) such that Sum_{k=a(n)..a(n+1)} 1/sqrt(k) > Pi.

Original entry on oeis.org

1, 5, 14, 28, 46, 69, 97, 130, 168, 211, 259, 311, 368, 430, 497, 569, 646, 728, 815, 907, 1004, 1105, 1211, 1322, 1438, 1559, 1685, 1816, 1952, 2093, 2239, 2390, 2546, 2706, 2871, 3041, 3216, 3396, 3581, 3771, 3966, 4166, 4371, 4581, 4796, 5016, 5240, 5469

Offset: 1

Benoit Cloitre, Aug 23 2002

Amiram Eldar, Table of n, a(n) for n = 1..600

Cf. A091476.

a[1] = 1; a[n_] := a[n] = Module[{k = a[n - 1], s = 0}, While[(s += 1/Sqrt[k]) < Pi, k++]; k]; Array[a, 50] (* Amiram Eldar, May 19 2022 *)

a(n) is asymptotic to Pi^2*n^2/4.

a(1) = 1 inserted by Amiram Eldar, May 19 2022

A175295 Decimal expansion of the integral of cos(Pix)log(x)/x^2 from x=1 to infinity.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 24 2010

			0.02991320398393497843930179...

G. C. Greubel, Table of n, a(n) for n = 0..10000
R. J. Mathar, Numerical evaluation of the oscillatory integral over exp(i*pi*x)*x^(1/x) between 1 and infinity, arXiv:0912.3844 [math.CA], 2009-2010, App. B.

evalf(1+Pi^2/2*( gamma+log(Pi)-1 ) -Pi^2*hypergeom([1/2,1/2,1], [3/2,3/2,3/2,2],-Pi^2/4)/2 ) ;

Join[{0}, RealDigits[ N[1/2*(Pi^2*(-2*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2, 3/2}, -Pi^2/4] + Log[Pi] + EulerGamma - 1) + 2*Pi*SinIntegral[Pi] - 2), 105]][[1]]] (* Jean-François Alcover, Nov 08 2012 *)
Join[{0},RealDigits[NIntegrate[Cos[Pi*x] Log[x]/x^2,{x,1,\[Infinity]}, WorkingPrecision->1000],10,120][[1]]] (* Harvey P. Dale, Nov 01 2017 *)

1+ A102753*( A053510 -1 + A001620 - 3F4(1/2,1/2,1; 3/2,3/2,3/2,2 ; -A091476) ) .

A261813 Decimal expansion of (Pi/4)^N*(N^N/N!)^2 for N = 3.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 19 2015

The general expression is a lower bound (due to H. Minkowski) on the discriminant of a number field of degree N.

The corresponding value for N = 2 matches A091476.

			9.8105797308761149773968028142000508257040952102995848563504202594...

B. Mazur, Algebraic Numbers, in The Princeton Companion to Mathematics, Editor T. Gowers, Princeton University Press, 2008, Section IV.1, page 330.

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A000796, A091476 (N=2).

n = 3; First@ RealDigits[N[(Pi/4)^n (n^n/n!)^2, 120]] (* Michael De Vlieger, Nov 19 2015 *)

PARI
```
N=3;(Pi/4)^N*(N^N/N!)^2
```

Equals 81*Pi^3/256.

A369880 Decimal expansion of sinh(Pi/2)/(Pi/2)^2.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 04 2024

			0.93268131478635101777369755780799023506619209387697...

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 424.

Jonathan M. Borwein and Peter B. Borwein, Strange series and high precision fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640. See p. 629, eq. (3.6); alternative link.

Cf. A000120, A019669, A060294, A096111, A091476, A185199, A258232, A308716, A367959.

RealDigits[Sinh[Pi/2]/(Pi/2)^2, 10, 120][[1]]

PARI
```
sinh(Pi/2)/(Pi/2)^2
```

Equals A060294 * A308716 = A308716 / A019669 = A185199 * A367959 = A367959 / A091476 = A367959 / A019669^2.

Equals Sum_{k>=0} (-1/16)^A000120(k)/D(k)^4, where D(k) = A096111(k-1) for k >= 1, and D(0) = 1 (Borwein and Borwein, 1992).

A145427 Decimal expansion of Sum_{n>=0} (n!/(n+3)!)^2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

			0.0299011002723396547...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.32.

Cf. A091476, A145426.

Maple
```
1/4*Pi^2-39/16 ;
```

RealDigits[Pi^2/4 - 39/16, 10, 120, -1][[1]] (* Amiram Eldar, Jun 19 2023 *)

Equals A091476 - 2.4375.

A225016 Decimal expansion of Pi^3/8.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 24 2013

			3.875784585037477521934539383387674400278161070735638461768067262975799364683...

Index entries for transcendental numbers

Cf. A000796, A002388, A019669, A091476, A091925.

Mathematica
```
RealDigits[Pi^3/8, 10, 100][[1]]
```

Pi^3/8 \\ Charles R Greathouse IV, Oct 01 2022

Equals Integral_{x>0} log(x)^2/(1+x^2) dx.
Equals Integral_{x=0..Pi/2} log(tan(x))^2 dx.
Equals Integral_{x=0..Pi/2} log(sin(x)^3)*log(sin(x))-(3*Pi/2)*log(2)^2 dx.
Equals (27/7) * Sum_{k>=0} binomial(2*k, k)/((2*k+1)^3*16^k);
Equals (27/7) * 4F3([1/2, 1/2, 1/2, 1/2], [3/2, 3/2, 3/2], 1/4), where pFq() is the generalized hypergeometric function.
From Amiram Eldar, Aug 21 2020: (Start)
Equals Integral_{x=0..oo} x^2/cosh(x) dx.
Equals 2 + Integral_{x=0..oo} x^2 * exp(-x) * tanh(x) dx. (End)
From Gleb Koloskov, Jun 15 2021: (Start)
Equals 2*Integral_{x=0..1} log(x)^2/(1+x^2) dx.
Equals 2*Integral_{x=1..oo} log(x)^2/(1+x^2) dx.
Equals 2*(-1)^n*Integral_{x=-1/e..0} W(n,x)*(1-W(n,x))*log(-W(n,x))^2/x/(1-W(n,x)^4) dx, where W=LambertW, for n=0 and n=-1. (End)

Offset corrected by Rick L. Shepherd, Jan 01 2014

A353127 Decimal expansion of Pi^2/4 - log(2).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Apr 24 2022

Limit of the series u where u(n) = 1/n when n is a square, and u(n) = (-1)^n/n otherwise.

			1.77425391971239434529139062851086121575292471744...

M. Lepez, Les Grands classiques de Mathématiques, Classes préparatoires scientifiques, Analyse, Exercices corrigés et commentés, MP-PC-PT, Bréal, 1995, Exercice 201, p. 29.

Cf. A091476 (Pi^2/4), A002162 (log(2)).

Maple
```
evalf(Pi^2/4 - log(2),100);
```

RealDigits[Pi^2/4 - Log[2], 10, 100][[1]] (* Amiram Eldar, Apr 24 2022 *)

Pi^2/4 - log(2) \\ Michel Marcus, Apr 24 2022

Equals A091476 - A002162.

Equals Sum_{k>=1} ( (-1)^k/k + 2/(2*k-1)^2 ).

cp's OEIS Frontend

A181411 a(n) = Sum_{k=0..n} C(n,k)*sigma(n+k) for n>=1.

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A363848 Decimal expansion of the arithmetic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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A363874 Decimal expansion of the harmonic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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A073347 a(1)=1; a(n+1) is the smallest integer > a(n) such that Sum_{k=a(n)..a(n+1)} 1/sqrt(k) > Pi.

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A175295 Decimal expansion of the integral of cos(Pi*x)*log(x)/x^2 from x=1 to infinity.

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A261813 Decimal expansion of (Pi/4)^N*(N^N/N!)^2 for N = 3.

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A369880 Decimal expansion of sinh(Pi/2)/(Pi/2)^2.

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A175295 Decimal expansion of the integral of cos(Pix)log(x)/x^2 from x=1 to infinity.