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A163930
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Decimal expansion of the higher order exponential integral constant gamma(2,1).
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3
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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
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OFFSET
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0,1
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COMMENTS
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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 =>0 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.
The values of the gamma(k,n) = G(k,n) coefficients can be determined with the Maple program.
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LINKS
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Table of n, a(n) for n=0..103.
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.
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FORMULA
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G(2,1) = gamma(2,1) = gamma^2/2+Pi^2/12.
G(k,n) = (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)) with G(0,n) = 1 for k => 0 and n => 1.
G(k,n+1) = G(k,n) -G(k-1,n)/n.
GF(n) = GAMMA(n-z)/GAMMA(n).
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EXAMPLE
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G(2,1) = 0.9890559953279725553953956515...
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MAPLE
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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;
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MATHEMATICA
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RealDigits[ N[ EulerGamma^2/2 + Pi^2/12, 105]][[1]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *)
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CROSSREFS
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Cf. A163931 (E(x,m,n)), A163927 (alpha(k,n)).
G(1,1) equals A001620 (gamma).
(gamma - G(1,n)) equals A001008(n-1)/A002805(n-1) for n =>2.
The structure of the G(k,n=1) formula leads to A036039.
Sequence in context: A175572 A021095 A090998 * A157371 A201994 A200003
Adjacent sequences: A163927 A163928 A163929 * A163931 A163932 A163933
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KEYWORD
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cons,easy,nonn
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AUTHOR
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Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Aug 17 2009
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STATUS
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approved
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