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A163927 Numerators of the higher order exponential integral constants alpha(k,4). 6
1, 49, 1897, 69553, 2515513, 90663937, 3264855049, 117543378001, 4231639039705, 152339702576545, 5484235568128681, 197432536935184369, 7107571838026381177, 255872590744254526273, 9211413307971174616393 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*Integral_{t>=x} E(t,m-1,n)/t^n 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 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(z,n) to generate the alpha(k,n) coefficients in each column.

Appears to equal the numerator of the multiple harmonic (star) sum Sum_{1 <= k_1 <= ... <= k_n <= 3} 1/(k_1^2*...*k_n^2)). If true, then a(n) = numerator( 3/2 - 3/(5*4^n) + 1/(10*9^n) ). - Peter Bala, Jan 31 2019

LINKS

Table of n, a(n) for n=0..14.

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

alpha(k,n) = (1/k) * Sum_{i=0..k-1} (Sum_{p=0..n-1}(p^(2*i-2*k))*alpha(i, n)) 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(z,n) = product((1-(z/k)^2)^(-1), k = 1..n-1) = (Pi*z/sin(Pi*z))/(Beta(n+z,n-z)/Beta(n,n)).

EXAMPLE

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

MAPLE

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-1);

# End program 1

coln:=4; nmax1 := 16; for n from 0 to nmax1 do A008955(n, 0):=1 end do: for n from 0 to nmax1 do A008955(n, n) := (n!)^2 end do: for n from 1 to nmax1 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^(2*m-2*n) od: GF(z, coln) := m!^2/f(m): GF(z, coln):=series(GF(z, coln), z, nmax1);

# End program 2

CROSSREFS

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(z,n) lead to A008955.

The denominators of a(1,n), n >= 2, lead to A007407.

Sequence in context: A014942 A260856 A065785 * A245036 A061615 A049682

Adjacent sequences:  A163924 A163925 A163926 * A163928 A163929 A163930

KEYWORD

easy,frac,nonn

AUTHOR

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

STATUS

approved

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Last modified September 20 09:04 EDT 2021. Contains 347579 sequences. (Running on oeis4.)