A081855 -id:A081855 - cp's OEIS frontend

A090998 Decimal expansion of lim_{k -> +-oo} k^2*(1 - Gamma(1+i/k)) where i^2 = -1 and Gamma is the Gamma function.

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

Offset: 0

Benoit Cloitre, Feb 29 2004

Limit_{k->oo} k*(1-Gamma(1+1/k)) = -Gamma'(1) = gamma = 0.577....

Decimal expansion of the higher-order exponential integral constant gamma(2,1). The higher-order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*Integral_{t=x..oo} (E(t,m-1,n)/t^n) dt for m >= 1 and n >= 1, with E(x,m=0,n) = exp(-x). The series expansions of the higher-order exponential integrals are dominated by the gamma(k,n) and the alpha(k,n) constants, see A163927. - Johannes W. Meijer and Nico Baken, Aug 13 2009

			G(2,1) = 0.9890559953279725553953956515...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No. 3, April 1987, pp. 209-211.

Cf. A163931 (E(x,m,n)), A163927 (alpha(k,n)), A001620 (gamma).
The structure of the G(k,n=1) formulas lead (replace gamma with G and Zeta with Z) to A036039. - Johannes W. Meijer and Nico Baken, Aug 13 2009
Cf. A081855.

SetDefaultRealField(RealField(100)); R:= RealField(); (6*EulerGamma(R)^2 + Pi(R)^2)/12; // G. C. Greubel, Feb 01 2019

ncol:=1; nmax:=5; kmax:=nmax; for n from 1 to nmax do G(0,n):=1 od: for n from 1 to nmax do for k from 1 to kmax do G(k,n):= expand((1/k)*((gamma-sum(p^(-1),p=1..n-1))* G(k-1,n)+sum((Zeta(k-i)-sum(p^(-(k-i)),p=1..n-1))*G(i,n),i=0..k-2))) od; od: for k from 0 to kmax do G(k,ncol):=G(k,ncol) od; # Johannes W. Meijer and Nico Baken, Aug 13 2009

RealDigits[(6*EulerGamma^2 + Pi^2)/12, 10, 104][[1]] (* Jean-François Alcover, Mar 04 2013 *)

default(realprecision, 100); (6*Euler^2 +Pi^2)/12 \\ G. C. Greubel, Feb 01 2019

numerical_approx((6*euler_gamma^2 + pi^2)/12, digits=100) # G. C. Greubel, Feb 01 2019

From Johannes W. Meijer and Nico Baken, Aug 13 2009: (Start)
G(2,1) = gamma(2,1) = gamma^2/2 + Pi^2/12.
G(k,n) = (1/k)*(gamma*G(k-1,n)) - (1/k)*Sum_{p=1..n-1} (p^(-1))* G(k-1,n) + (1/k) * Sum_{i=0..k-2} (Zeta(k-i) * G(i,n))  - (1/k)*Sum_{i=0..k-2}(Sum_{p=1..n-1} (p^(i-k)) * G(i,n)) with G(0,n) = 1 for k >= 0 and n >= 1.
G(k,n+1) = G(k,n) - G(k-1,n)/n.
GF(z,n) = GAMMA(n-z)/GAMMA(n).
(gamma - G(1,n)) = A001008(n-1)/A002805(n-1) for n >= 2. (End)
Equals A081855/2. - Hugo Pfoertner, Mar 12 2024

A261509 Decimal expansion of -Gamma'''(1).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Aug 22 2015

			5.4448744564853177340993610041376506895716686944353825656479...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), p. 223-232.
Eric Weisstein's World of Mathematics, Gamma Function.
Wikipedia, Gamma function.

Cf. A001620, A081855, A002117.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^3 + (EulerGamma(R)*Pi(R)^2)/2 + 2*Evaluate(L,3); // G. C. Greubel, Aug 30 2018

RealDigits[EulerGamma^3 + (EulerGamma*Pi^2)/2 + 2*Zeta[3], 10, 120][[1]]

default(realprecision, 100); Euler^3 + Euler*Pi^2/2 + 2*zeta(3) \\ G. C. Greubel, Aug 30 2018

From Amiram Eldar, Aug 06 2020: (Start)
Equals gamma^3 + gamma*Pi^2/2 + 2*zeta(3).
Equals -Integral_{x=0..oo} exp(-x)*log(x)^3 dx. (End)

A291486 Decimal expansion of Gamma''''(1).

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Aug 24 2017

			23.56147408402560449607312705652442040865376831336316996971897893425256...

G. C. Greubel, Table of n, a(n) for n = 2..10000

Cf. A000796 (Pi), A001620 (EulerGamma), A002117 (zeta(3)), A081855, A261509.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^4 + EulerGamma(R)^2*Pi(R)^2 + 8*EulerGamma(R)*Evaluate(L,3) + 3*Pi(R)^4/20; // G. C. Greubel, Sep 07 2018

c:= subs(x=1.0, diff(GAMMA(x), x$4)):
evalf(c, 120);  # Alois P. Heinz, Jul 01 2023

Mathematica
```
RealDigits[Gamma''''[1], 10, 111][[1]]
```

default(realprecision, 100); Euler^4 + Euler^2*Pi^2 + 8*Euler*zeta(3) + 3*Pi^4/20 \\ G. C. Greubel, Sep 07 2018

Equals EulerGamma^4 + EulerGamma^2*Pi^2 + 8*EulerGamma*Zeta(3) + 3*Pi^4/20.

Equals Integral_{x=0..oo} exp(-x)*log(x)^4 dx. - Amiram Eldar, Aug 06 2020

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A090998 Decimal expansion of lim_{k -> +-oo} k^2*(1 - Gamma(1+i/k)) where i^2 = -1 and Gamma is the Gamma function.

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A261509 Decimal expansion of -Gamma'''(1).

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A291486 Decimal expansion of Gamma''''(1).

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Magma

Maple

Mathematica

PARI

Formula