0,2

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 alpha(k,n) and the gamma(k,n) constants, see A163930.

The first Maple program uses the alpha(k,n) formula and the second the GF(n) to generate the alpha(k,n) coefficients in each column.

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.

alpha(k,n) = (1/k)*sum(sum(p^(-2*(k-i)),p = 0..n-1)*alpha(i, n), i = 0..k-1) with alpha(0,n) = 1, k => 0 and n => 1.

alpha(k,n) = alpha(k,n+1) -alpha(k-1,n+1)/n^2

GF(n) = product((1-(z/k)^2)^(-1), k = 1..n-1) = (Pi*z/sin(Pi*z))/(Beta(n+z,n-z)/Beta(n,n))

a(k=0,n=4) = 1, a(k=1,4) = 49/36, a(k=2,4) = 1897/1296, a(k=3,4) = 69553/46656.

restart; coln:=4; nmax:=15; kmax:=nmax: k:=0: for n from 1 to nmax do alpha(k, n):=1 od: for k from 1 to kmax do for n from 1 to nmax do alpha(k, n) := (1/k)*sum(sum(p^(-2*(k-i)), p=0..n-1)*alpha(i, n), i=0..k-1) od; od: seq(alpha(k, coln), k=0..nmax);

restart; coln:=4; nmax:=8: mmax:=nmax: for n from 0 to nmax do A008955(n, 0):=1 end do: for n from 0 to nmax do A008955(n, n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n, m):= A008955(n-1, m-1)*n^2+A008955(n-1, m) end do: end do: m:=coln-1: f(m):=0: for n from 0 to m do f(m):=f(m)+(-1)^(n+m)*A008955(m, n)*z^(m-n) od: GF(coln):=m!^2/f(m): GF(coln):=series(GF(coln), z, nmax);

Cf. A163931 (E(x, m, n)), A163930 (gamma(k, n)).

Cf. A163928 y A163929.

a(k,1) = A000007(k)

a(k,2) = A000012(k) = 1^k.

a(k,3) = A002450(k+1)/A000302(k) with A000302(k) = 4^k.

a(k,4) = A163927(k)/A009980(k) with A009980(k) = 36^k.

The GF(n) lead to A008955.

Sequence in context: A145848 A014942 A065785 this_sequence A061615 A049682 A120999

Adjacent sequences: A163924 A163925 A163926 this_sequence A163928 A163929 A163930

easy,frac,nonn

Johannes W. Meijer & Nico Baken (meijgia(AT)hotmail.com and n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Aug 17 2009