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

Author

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

Keywords

• nonn
• frac
• easy

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

Examples

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

Crossrefs

Cf. A163929 (denominators).

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

Cf. A007406, A007407.

Programs

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);

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.