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Duplicate of A090998
1

%I #17 Jan 25 2023 06:23:51

%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 Duplicate of A090998

%C The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*int(E(t,m-1,n)/t^n, t=x..infinity) for m>=1 and n>=1, with E(x,m=0,n) = exp(-x).

%C The series expansions of the higher order exponential integrals are dominated by the gamma(k,n) and the alpha(k,n) constants, see A163927.

%C The values of the gamma(k,n) = G(k,n) coefficients can be determined with the Maple program.

%H G. C. Greubel, <a href="/A163930/b163930.txt">Table of n, a(n) for n = 0..5000</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 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).

%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;

%t RealDigits[ N[ EulerGamma^2/2 + Pi^2/12, 105]][[1]] (* _Jean-François Alcover_, Nov 07 2012, from 1st formula *)

%Y Cf. A163931 (E(x,m,n)), A163927 (alpha(k,n)).

%Y G(1,1) equals A001620 (gamma).

%Y (gamma - G(1,n)) equals A001008(n-1)/A002805(n-1) for n>=2.

%Y The structure of the G(k,n=1) formulas lead (replace gamma by G and Zeta by Z) to A036039.

%K dead

%O 0,1

%A _Johannes W. Meijer_ and _Nico Baken_, Aug 13 2009, Aug 17 2009