A352593 Denominator values occurring in formulas for the n-th integration of the Lambert W function.

1, 8, 648, 82944, 1296000000, 69984000000, 403443833184000000, 26440095051546624000000, 42154051662866968215552000000, 263462822892918551347200000000000, 826859199578154686310659783668531200000000000

Offset

1,2

Links

Table of n, a(n) for n=1..11.

Example

If we use J(n, f(x)) notation for the n-th integration, then we can find the denominator
  J(1, W(x)) = (x/W(x))*(W(x)^2 - W(x) + 1) + c
  J(2, W(x)) = (x^2/(8*W(x)^2))*(4*W(x)^3 - 6*W(x)^2 + 6*W(x) + 1) + k*x + c
  J(3, W(x)) = (x^3/(648*W(x)^3))*(108*W(x)^4 - 198*W(x)^3 + 198*W(x)^2 + 57*W(x) + 8) + h*x^2/2 + k*x + c
  ...
where c, k, h are constants.

Mathematica

max = 10; Table[Denominator[Together[Rest[NestList[Integrate[#, x] &, LambertW[x], max]]]][[k]] / ProductLog[x]^k, {k, 1, max}] (* Vaclav Kotesovec, Apr 14 2022 *)

Crossrefs

Cf. A352592.

Sequence in context: A196588 A197045 A258385 * A267968 A253268 A235368

Adjacent sequences: A352590 A352591 A352592 * A352594 A352595 A352596

Keyword

nonn

Author

Lukáš Backa, Mar 21 2022

Extensions

More terms from Vaclav Kotesovec, Apr 14 2022

Status

approved