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 A163930 Decimal expansion of the higher order exponential integral constant gamma(2,1). 3
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. LINKS 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. FORMULA 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). EXAMPLE G(2,1) = 0.9890559953279725553953956515... MAPLE 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; MATHEMATICA RealDigits[ N[ EulerGamma^2/2 + Pi^2/12, 105]][[1]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *) CROSSREFS 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 KEYWORD cons,easy,nonn AUTHOR Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Aug 17 2009 STATUS approved

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Last modified May 22 19:14 EDT 2013. Contains 225562 sequences.