A097667 -id:A097667 - cp's OEIS frontend

A097680 E.g.f.: (1/(1-x^5))exp( 5sum_{i>=0} x^(5i+1)/(5i+1) ) for an order-5 linear recurrence with varying coefficients.

1, 5, 25, 125, 625, 3245, 19825, 162125, 1650625, 17703125, 186644425, 2032320125, 25569960625, 382772328125, 6166860390625, 98093486946125, 1555728351450625, 26765871718953125, 527380555479765625, 11241893092061328125

Offset: 0

Paul D. Hanna, Sep 01 2004

nonn

Limit_{n->inf} n*n!/a(n) = 5*c = 0.2247091438... where c = 5*exp(psi(1/5)+EulerGamma) = 0.0449418287...(A097667) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

			The sequence {1, 5, 25/2!, 125/3!, 625/4!, 3245/5!, 19825/6!, 162125/7!,...} 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. A097667, A097677-A097679, A097681-A097682.

{a(n)=n!*polcoeff(1/(1-x^5)*exp(5*sum(i=0,n,x^(5*i+1)/(5*i+1)))+x*O(x^n),n)}

a(n)=if(n<0,0,if(n==0,1,5*a(n-1)+if(n<5,0,n!/(n-5)!*a(n-5))))

For n>=5: a(n) = 5*a(n-1) + n!/(n-5)!*a(n-5); for n<5: a(n)=5^n. E.g.f.: B(x)*exp(C(x)) where B(x) = 1/(1-x^5)/(1-x)*(1+phi*x+x^2)^(phi/2)/(1-x/phi+x^2)^(1/phi/2) and C(x) = 5^(1/4)*sqrt(phi)*atan(5^(1/4)*sqrt(phi)*x/(2-x/phi)) + 5^(1/4)/sqrt(phi)*atan(5^(1/4)/sqrt(phi)*x/(2+phi*x)) and where phi=(sqrt(5)+1)/2.

A097666 Decimal expansion of the constant 4*exp(psi(3/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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-4 linear recursions with varying coefficients (see A097679 for example).

			c = 2.40523869048267582773651783335191656319508543733226747001040...

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-A097665, A097667-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(Pi(R)/2)/2; // G. C. Greubel, Sep 07 2018

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

PARI
```
4*exp(psi(3/4)+Euler)
```

c = 1/2*exp(Pi/2).

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

cp's OEIS Frontend

A097680 E.g.f.: (1/(1-x^5))*exp( 5*sum_{i>=0} x^(5*i+1)/(5*i+1) ) for an order-5 linear recurrence with varying coefficients.

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A097666 Decimal expansion of the constant 4*exp(psi(3/4) + 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.

Views

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Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A097680 E.g.f.: (1/(1-x^5))exp( 5sum_{i>=0} x^(5i+1)/(5i+1) ) for an order-5 linear recurrence with varying coefficients.