A352592 Coefficients occurring in the polynomials of the n-th integration of the principal branch of the Lambert W function.

1, -1, 1, 4, -6, 6, 1, 108, -198, 198, 57, 8, 3456, -7200, 7200, 2760, 700, 81, 10800000, -24660000, 24660000, 11397000, 3974300, 850905, 82944

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).

Links

Table of n, a(n) for n=1..25.

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

Cf. A352593.

Sequence in context: A324942 A199161 A329217 * A119858 A130753 A021686

Adjacent sequences: A352589 A352590 A352591 * A352593 A352594 A352595

Keyword

tabf,sign,more

Author

Lukáš Backa, Mar 21 2022

Status

approved