A097680 -id:A097680 - cp's OEIS frontend

A097679 E.g.f.: (1/(1-x^4))exp( 4Sum_{i>=0} x^(4i+1)/(4i+1) ) for an order-4 linear recurrence with varying coefficients.

1, 4, 16, 64, 280, 1600, 12160, 102400, 880000, 8358400, 94720000, 1189888000, 15213952000, 204285952000, 3092697088000, 51351519232000, 869951500288000, 15148619579392000, 287722152460288000, 5927812334878720000

Offset: 0

Paul D. Hanna, Sep 01 2004

nonn

Lim_{n->inf} n*n!/a(n) = 4*c = 0.4157591527... where c = 4*exp(psi(1/4)+EulerGamma) = 0.1039397881...(A097665) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

			The sequence {1, 4, 16/2!, 64/3!, 280/4!, 1600/5!, 12160/6!, 102400/7!,...} is generated by a recursion described by _Benoit Cloitre_'s generalized Euler-Gauss formula for the Gamma function (see Cloitre link).

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Eric Weisstein's World of Mathematics, Digamma Function.

Cf. A097665, A097677, A097678, A097680, A097681, A097682.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)/(1-x^4)/(1-x)*Exp(2*Arctan(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 29 2018

Range[0, 20]! CoefficientList[ Series[ E^(4Sum[x^(4k + 1)/(4k + 1), {k, 0, 150}])/(1 - x^4), {x, 0, 20}], x] (* Robert G. Wilson v, Sep 03 2004 *)

{a(n)=n!*polcoeff(1/(1-x^4)*exp(4*sum(i=0,n,x^(4*i+1)/(4*i+1)))+x*O(x^n),n)}

a(n)=if(n<0,0,if(n==0,1,4*a(n-1)+if(n<4,0,n!/(n-4)!*a(n-4))))

For n>=4: a(n) = 4*a(n-1) + n!/(n-4)!*a(n-4); for n<4: a(n)=4^n.

E.g.f.: (1+x)/(1-x^4)/(1-x)*exp(2*atan(x)).

A097681 E.g.f.: (1/(1-x^6))exp( 6sum_{i>=0} x^(6i+1)/(6i+1) ) for an order-6 linear recurrence with varying coefficients.

Original entry on oeis.org

1, 6, 36, 216, 1296, 7776, 47376, 314496, 2612736, 28740096, 368395776, 4796983296, 60300205056, 750367328256, 10151357239296, 164475953381376, 3110937349718016, 61410199093641216, 1174438559356747776

Offset: 0

Paul D. Hanna, Sep 01 2004

nonn

Limit_{n->inf} n*n!/a(n) = 6*c = 0.1140186893... where c = 6*exp(psi(1/6)+EulerGamma) = 0.0190031148...(A097671) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

			The sequence {1, 6, 36/2!, 216/3!, 1296/4!, 7776/5!, 47376/6!,...} is generated by a recursion described by Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link).

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Eric Weisstein's World of Mathematics, Digamma Function.

Cf. A097671, A097677-A097680, A097682-A097682.

{a(n)=n!*polcoeff(1/(1-x^6)*exp(6*sum(i=0,n,x^(6*i+1)/(6*i+1)))+x*O(x^n),n)}

a(n)=if(n<0,0,if(n==0,1,6*a(n-1)+if(n<6,0,n!/(n-6)!*a(n-6))))

For n>=6: a(n) = 6*a(n-1) + n!/(n-6)!*a(n-6); for n<6: a(n)=6^n. E.g.f.: 1/(1-x^6)*(1+x)/(1-x)*sqrt((1+x+x^2)/(1-x+x^2))* exp(sqrt(3)*atan(sqrt(3)*x/(1-x^2))).

A097667 Decimal expansion of the constant 5*exp(psi(1/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 0.04494182877920882060846739642766520340238594371059869805861...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097666, A097668-A097676.

RealDigits[ GoldenRatio^(-Sqrt[5]/2)/5^(1/4)*E^(-Pi/2*Sqrt[1 + 2/Sqrt[5]]), 10, 104][[1]] (* Robert G. Wilson v, Aug 27 2004 *)
Join[{0}, RealDigits[N[5*Exp[PolyGamma[1/5] + EulerGamma], 120], 10, 100][[1]]] (* G. C. Greubel, Dec 31 2016 *)

PARI
```
5*exp(psi(1/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(-sqrt(5)/2)/5^(1/4)*exp(-Pi/2*sqrt(1+2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A097668 Decimal expansion of the constant 5*exp(psi(2/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 0.68747420699080196070816422133398475499777353078320593237327...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097667, A097669-A097676.

RealDigits[ GoldenRatio^(Sqrt[5]/2)/5^(1/4)*E^(-Pi/2Sqrt[1 - 2/Sqrt[5]]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
5*exp(psi(2/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(sqrt(5)/2)/5^(1/4)*exp(-Pi/2*sqrt(1-2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A097669 Decimal expansion of the constant 5*exp(psi(3/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 1.90795953254354252255333813972952036908516068359082968228223...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097668, A097670-A097676.

RealDigits[ GoldenRatio^(Sqrt[5]/2)/5^(1/4)*E^(Pi/2Sqrt[1 - 2/Sqrt[5]]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
5*exp(psi(3/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(sqrt(5)/2)/5^(1/4)*exp(Pi/2*sqrt(1-2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A097670 Decimal expansion of the constant 5*exp(psi(4/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 3.39276427892784980770475505655447128392740109258608442234780...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097669, A097671-A097676.

RealDigits[ GoldenRatio^(-Sqrt[5]/2)/5^(1/4)*E^(Pi/2Sqrt[1 + 2/Sqrt[5]]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
5*exp(psi(4/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(-sqrt(5)/2)/5^(1/4)*exp(Pi/2*sqrt(1+2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

cp's OEIS Frontend

A097679 E.g.f.: (1/(1-x^4))*exp( 4*Sum_{i>=0} x^(4*i+1)/(4*i+1) ) for an order-4 linear recurrence with varying coefficients.

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A097681 E.g.f.: (1/(1-x^6))*exp( 6*sum_{i>=0} x^(6*i+1)/(6*i+1) ) for an order-6 linear recurrence with varying coefficients.

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A097667 Decimal expansion of the constant 5*exp(psi(1/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097668 Decimal expansion of the constant 5*exp(psi(2/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097669 Decimal expansion of the constant 5*exp(psi(3/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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Programs

Mathematica

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Extensions

A097670 Decimal expansion of the constant 5*exp(psi(4/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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Author

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Comments

Examples

References

Links

Crossrefs

A097679 E.g.f.: (1/(1-x^4))exp( 4Sum_{i>=0} x^(4i+1)/(4i+1) ) for an order-4 linear recurrence with varying coefficients.

A097681 E.g.f.: (1/(1-x^6))exp( 6sum_{i>=0} x^(6i+1)/(6i+1) ) for an order-6 linear recurrence with varying coefficients.