A163928 Numerators of the higher order exponential integral constants alpha(2,n).

0, 1, 21, 1897, 32197, 20881861, 7139587, 17462165587, 283355376967, 69621962857381, 70246946681461, 1036088178214798501, 1042504974775473001, 29931734181763981573561, 4295332813075795410223, 4312254507400142830831

Offset

1,3

Comments

See A163927 for information about the alpha(k,n) constants.

Apart from a difference of offset, alpha(2,n) appears to be the multiple harmonic (star) sum Sum_{j = 1..n} 1/j^2 Sum_{k = 1..j} 1/k^2, which has the initial values [1, 21/16, 1897/1296, 32197/20736, 20881861/12960000, 7139587/4320000, ...]. - Peter Bala, Jan 31 2019

Links

Table of n, a(n) for n=1..16.

Formula

alpha(k,n) = (1/k)*Sum_{i=0..k-1} (Sum_{p=0..n-1} p^(-2*(k-i))*alpha(i, n) with alpha(0,n) = 1, with k = 2 and n >= 1. alpha(1,n) = A007406(n-1)/A007407(n-1) for n >= 2.

Example

alpha(k=2,n=1) = 0, alpha(k=2,2) = 1, alpha(k=2,3) = 21/16, alpha(k=2,4) = 1897/1296.

Maple

nmax:=17; rowk:=2; 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(rowk, n), n=1..nmax);

Crossrefs

Cf. A163929 (denominators).

Cf. A163927 (alpha(k,n)) and A090998 (gamma(k,n)).

Cf. A007406, A007407.

Sequence in context: A232949 A305145 A296686 * A358803 A243685 A221122

Adjacent sequences: A163925 A163926 A163927 * A163929 A163930 A163931

Keyword

nonn,frac,easy

Author

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

Status

approved