A097679 -id:A097679 - cp's OEIS frontend

A097663 Decimal expansion of the constant 3*exp(psi(1/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

2, 3, 3, 1, 1, 9, 0, 9, 3, 1, 8, 4, 5, 6, 4, 1, 1, 7, 3, 0, 5, 3, 7, 5, 6, 2, 3, 2, 6, 5, 4, 4, 2, 8, 9, 5, 7, 4, 4, 6, 0, 8, 5, 8, 7, 0, 2, 5, 9, 2, 4, 5, 6, 4, 1, 4, 0, 9, 6, 0, 0, 7, 8, 7, 5, 6, 1, 6, 8, 2, 8, 5, 3, 1, 1, 5, 3, 1, 7, 4, 6, 3, 3, 5, 1, 1, 2, 2, 5, 5, 6, 6, 9, 4, 0, 6, 7, 7, 7, 0, 3, 3, 8, 9, 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-3 linear recursions with varying coefficients (see A097677 for example).

			0.23311909318456411730537562326544289574460858702592456414096...

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..5000
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. A001620, A047787, A097664, A097665-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(-Pi(R)/Sqrt(12))/Sqrt(3); // G. C. Greubel, Sep 07 2018

RealDigits[1/Sqrt[3]*E^(-Pi/Sqrt[12]), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
3*exp(psi(1/3)+Euler)
```

Equals exp(-Pi/sqrt(12))/sqrt(3).

More terms from Robert G. Wilson v, Aug 28 2004

Offset corrected by R. J. Mathar, Feb 05 2009

A097676 Decimal expansion of the constant 8*exp(psi(7/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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-8 linear recursions with varying coefficients (see A097682 for example).

			c = 6.37663248941667785500176259382510790626743532678646216767306...

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..5000
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-A097675.

SetDefaultRealField(RealField(100)); R:= RealField();
(1+Sqrt(2))^(-Sqrt(2))/2*Exp(Pi(R)/2*(1+Sqrt(2))); // G. C. Greubel, Sep 07 2018

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

PARI
```
8*exp(psi(7/8)+Euler)
```

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

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

A097682 E.g.f.: (1/(1-x^8))exp( 8sum_{i>=0} x^(8i+1)/(8i+1) ) for an order-8 linear recurrence with varying coefficients.

Original entry on oeis.org

1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16817536, 137443328, 1215668224, 13131579392, 186802241536, 3194809745408, 57299125141504, 1002518381330432, 16747075923705856, 268695698674024448, 4294396462470529024

Offset: 0

Paul D. Hanna, Sep 01 2004

nonn

Limit_{n->inf} n*n!/a(n) = 8*c = 0.0259289826... where c = 8*exp(psi(1/8)+EulerGamma) = 0.0032411228...(A097673) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

			The sequence {1, 8, 64/2!, 512/3!, 4096/4!, 32768/5!, 262144/6!,...} 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. A097673, A097677-A097681.

{a(n)=n!*polcoeff(1/(1-x^8)*exp(8*sum(i=0,n,x^(8*i+1)/(8*i+1)))+x*O(x^n),n)}

a(n)=if(n<0,0,if(n==0,1,8*a(n-1)+if(n<8,0,n!/(n-8)!*a(n-8))))

For n>=8: a(n) = 8*a(n-1) + n!/(n-8)!*a(n-8); for n<8: a(n)=8^n. E.g.f.: 1/(1-x^8)*(1+x)/(1-x)* ((1+sqrt(2)*x+x^2)/(1-sqrt(2)*x+x^2))^(1/sqrt(2))* exp(sqrt(2)*atan(sqrt(2)*x/(1-x^2))+2*atan(x)).

A097665 Decimal expansion of the constant 4*exp(psi(1/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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

			c = 0.10393978817538095427347780991748938501693892081588480403756...

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.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.

Cf. A097663-A097664, A097666-A097676.

RealDigits[1/2*E^(-Pi/2), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
4*exp(psi(1/4)+Euler)
```

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

More terms from Robert G. Wilson v, Aug 28 2004

A097677 E.g.f.: (1/(1-x^3))exp( 3sum_{i>=0} x^(3i+1)/(3i+1) ) for an order-3 linear recurrence with varying coefficients.

Original entry on oeis.org

1, 3, 9, 33, 171, 1053, 7119, 57267, 525609, 5164803, 56726649, 690532857, 8889138531, 124010345277, 1880154795519, 29907812576187, 506398197859281, 9190226159295363, 173999328850897641, 3466197108906552657

Offset: 0

Paul D. Hanna, Sep 01 2004

nonn

Limit_{n->inf} n*n!/a(n) = 3*c = 0.6993572795... where c = 3*exp(psi(1/3)+EulerGamma) = 0.2331190931...(A097663) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function.

			The sequence {1, 3, 9/2!, 33/3!, 171/4!, 1053/5!, 7119/6!, 57267/7!,...} 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. A097663, A097678-A097682.

{a(n)=n!*polcoeff(1/(1-x^3)*exp(3*sum(i=0,n,x^(3*i+1)/(3*i+1)))+x*O(x^n),n)}

a(n)=if(n<0,0,if(n==0,1,3*a(n-1)+if(n<3,0,n!/(n-3)!*a(n-3))))

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

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.

Original entry on oeis.org

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.

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))).

A097664 Decimal expansion of the constant 3*exp(psi(2/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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-3 linear recursions with varying coefficients (see A097678 for example).

			c = 1.42988430840123420566179042477513809656498236767564464887634...

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.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.

Cf. A097663, A097665-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); (1/Sqrt(3))*Exp(Pi(R)/Sqrt(12)); // G. C. Greubel, Aug 27 2018

RealDigits[1/Sqrt[3]*E^(Pi/Sqrt[12]), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
3*exp(psi(2/3)+Euler)
```

c = 1/sqrt(3)*exp(Pi/sqrt(12)).

More terms from Robert G. Wilson v, Aug 28 2004

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

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

cp's OEIS Frontend

A097663 Decimal expansion of the constant 3*exp(psi(1/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097676 Decimal expansion of the constant 8*exp(psi(7/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097682 E.g.f.: (1/(1-x^8))*exp( 8*sum_{i>=0} x^(8*i+1)/(8*i+1) ) for an order-8 linear recurrence with varying coefficients.

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A097665 Decimal expansion of the constant 4*exp(psi(1/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097677 E.g.f.: (1/(1-x^3))*exp( 3*sum_{i>=0} x^(3*i+1)/(3*i+1) ) for an order-3 linear recurrence with varying coefficients.

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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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Crossrefs

A097682 E.g.f.: (1/(1-x^8))exp( 8sum_{i>=0} x^(8i+1)/(8i+1) ) for an order-8 linear recurrence with varying coefficients.

A097677 E.g.f.: (1/(1-x^3))exp( 3sum_{i>=0} x^(3i+1)/(3i+1) ) for an order-3 linear recurrence with varying coefficients.

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.

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.