%I #18 Jan 25 2023 06:08:31

%S 0,1,21,1897,32197,20881861,7139587,17462165587,283355376967,

%T 69621962857381,70246946681461,1036088178214798501,

%U 1042504974775473001,29931734181763981573561,4295332813075795410223,4312254507400142830831

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

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

%C 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

%F 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.

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

%p 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);

%Y Cf. A163929 (denominators).

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

%Y Cf. A007406, A007407.

%K nonn,frac,easy

%O 1,3

%A _Johannes W. Meijer_ & _Nico Baken_, Aug 13 2009, Aug 17 2009