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%I #24 Mar 12 2024 09:53:39
%S 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,
%T 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,
%U 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
%N Decimal expansion of lim_{k -> +-oo} k^2*(1 - Gamma(1+i/k)) where i^2 = -1 and Gamma is the Gamma function.
%C Limit_{k->oo} k*(1-Gamma(1+1/k)) = -Gamma'(1) = gamma = 0.577....
%C 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
%H G. C. Greubel, <a href="/A090998/b090998.txt">Table of n, a(n) for n = 0..10000</a>
%H J. W. Meijer and N. H. G. Baken, <a href="https://doi.org/10.1016/0167-7152(87)90041-1">The Exponential Integral Distribution</a>, Statistics and Probability Letters, Volume 5, No. 3, April 1987, pp. 209-211.
%F From _Johannes W. Meijer_ and _Nico Baken_, Aug 13 2009: (Start)
%F G(2,1) = gamma(2,1) = gamma^2/2 + Pi^2/12.
%F 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.
%F G(k,n+1) = G(k,n) - G(k-1,n)/n.
%F GF(z,n) = GAMMA(n-z)/GAMMA(n).
%F (gamma - G(1,n)) = A001008(n-1)/A002805(n-1) for n >= 2. (End)
%F Equals A081855/2. - _Hugo Pfoertner_, Mar 12 2024
%e G(2,1) = 0.9890559953279725553953956515...
%p 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
%t RealDigits[(6*EulerGamma^2 + Pi^2)/12, 10, 104][[1]] (* _Jean-François Alcover_, Mar 04 2013 *)
%o (PARI) default(realprecision, 100); (6*Euler^2 +Pi^2)/12 \\ _G. C. Greubel_, Feb 01 2019
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (6*EulerGamma(R)^2 + Pi(R)^2)/12; // _G. C. Greubel_, Feb 01 2019
%o (Sage) numerical_approx((6*euler_gamma^2 + pi^2)/12, digits=100) # _G. C. Greubel_, Feb 01 2019
%Y Cf. A163931 (E(x,m,n)), A163927 (alpha(k,n)), A001620 (gamma).
%Y 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
%Y Cf. A081855.
%K cons,nonn
%O 0,1
%A _Benoit Cloitre_, Feb 29 2004
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