OFFSET
1,4
COMMENTS
The sequence can be written in the shape of a triangle T(n, k), where the n-th row represents the coefficients for the n-th integration of Lambert W function
n\k| 0 1 2 3 4 5
---+----------------------------------------------------
1 | 1 -1 1 0 0 0
2 | 4 -6 6 1 0 0
3 | 108 -198 198 57 8 0
4 | 3456 -7200 7200 2760 700 81
5 | 10800000 -24660000 24660000 11397000 3974300 850905
It appears that T(n, 1) = -T(n, 2).
EXAMPLE
If we use the notation J(n, f(x)) for the n-th integration, we can rewrite the triangle into expressions
J(1, W(x)) = (x/W(x))*(W(x)^2 - W(x) + 1) + c,
J(2, W(x)) = (x^2/(8*W(x)^4))*(4*W(x)^3 - 6*W(x)^2 + 6W(x) + 1) + k*x + c,
J(3, W(x)) = (x^3/(648*W(x)^4))*(108*W(x)^4 - 198*W(x)^3 + 198*W(x)^2 + 57*W(x) + 8) + (x^2/2)*h + k*x + c,
...
where c, k, h are constants.
MATHEMATICA
max = 5; Flatten[Table[Reverse[Rest[CoefficientList[Numerator[Together[ NestList[Integrate[#, x] &, LambertW[x], max]]], ProductLog[x]]][[k]]]/x^k, {k, 1, max}]] (* Vaclav Kotesovec, Apr 14 2022 *)
CROSSREFS
KEYWORD
tabf,sign,more
AUTHOR
Lukáš Backa, Mar 21 2022
STATUS
approved