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Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:

$\operatorname{li}^{-1}(n)=n\sum _{i=1} a_i(-1)^{i+1} W_{-1}\left(-e/n\right){}^{2-i} $

where $a_i$ begins: $\small{-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, \dots}$

where the next one $\small{\approx 1465235 + \epsilon, \ \epsilon =-1/12?)}$.

The above should be accurate to $o(1/\log(n)^{10})$ - is this correct? Is there a nice way to obtain the coefficients other than by computation?

For convenience:

f[n_] := With[
{a = {-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, 17582819/12}}, 
n Sum[(-1)^(1 + k) a[[k]] ProductLog[-1, -E/n]^(2 - k), {k, 1, 11}]];
g[n_] := Quiet[x /. FindRoot[LogIntegral@x == n, {x, N[n Log[n], 200]}, 
WorkingPrecision -> 200]];

eg Abs@N[1 - f@#/g@#] &[10^10^7]

Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:

$\operatorname{li}^{-1}(n)=n\sum _{i=1} a_i(-1)^{i+1} W_{-1}\left(-e/n\right){}^{2-i} $

where $a_i$ begins: $\small{-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, \dots}$

where the next one $\small{\approx 1465235 + \epsilon, \ \epsilon =-1/12?)}$.

The above should be accurate to $o(1/\log(n)^{10})$ - is this correct? Is there a nice way to obtain the coefficients other than by computation?

For convenience:

f[n_] := With[
{a = {-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, 17582819/12}}, 
n Sum[(-1)^(1 + k) a[[k]] ProductLog[-1, -E/n]^(2 - k), {k, 1, 11}]];
g[n_] := Quiet[x /. FindRoot[LogIntegral@x == n, {x, N[n Log[n], 200]}, 
WorkingPrecision -> 200]]; 

Abs@Log@N[1/Log[#]^10] &[10^10^7]
Abs@Log@N[1 - f@#/g@#] &[10^10^7]

Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:

$\operatorname{li}^{-1}(n)=n\sum _{i=1} a_i(-1)^{i+1} W_{-1}\left(-e/n\right){}^{2-i} $

where $a_i$ begins: $\small{-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, \dots}$

where the next one $\small{\approx 1465235 + \epsilon, \ \epsilon =-1/12?)}$.

The above should be accurate to $o(1/\log(n)^{10})$ - is this correct? Is there a nice way to obtain the coefficients other than by computation?

For convenience:

f[n_] := With[
{a = {-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, 17582819/12}}, 
n Sum[(-1)^(1 + k) a[[k]] ProductLog[-1, -E/n]^(2 - k), {k, 1, 11}]];
g[n_] := Quiet[x /. FindRoot[LogIntegral@x == n, {x, N[n Log[n], 200]}, 
WorkingPrecision -> 200]];

Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:

$\operatorname{li}^{-1}(n)=n\sum _{i=1} a_i(-1)^{i+1} W_{-1}\left(-e/n\right){}^{2-i} $

where $a_i$ begins: $\small{-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, \dots}$

where the next one $\small{\approx 1465235 + \epsilon, \ \epsilon =-1/12?)}$.

The above should be accurate to $o(1/\log(n)^{10})$ - is this correct? Is there a nice way to obtain the coefficients other than by computation?

For convenience:

f[n_] := With[
{a = {-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, 17582819/12}}, 
n Sum[(-1)^(1 + k) a[[k]] ProductLog[-1, -E/n]^(2 - k), {k, 1, 11}]];
g[n_] := Quiet[x /. FindRoot[LogIntegral@x == n, {x, N[n Log[n], 200]}, 
WorkingPrecision -> 200]];

eg Abs@N[1 - f@#/g@#] &[10^10^7]