A340474 a(n) = n! [x^n] LW(T(x)), where T(x) = -W(-x) Euler's tree function, W(x) is the Lambert W function, and LW(x) = W(-W(x))/(-W(x)) (A340473).

1, 1, 3, 22, 209, 2756, 43717, 839686, 18581425, 470707192, 13352676101, 420875581754, 14566375690297, 549877190829604, 22472783629465093, 989043215802778966, 46631075599107558113, 2345376059569552767344, 125350843842721213505029, 7095169059445749303612946

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

a(n) ~ n^(n-1) * exp(1/2 - n*exp(-1) + n*exp(-1+exp(-1))) / (sqrt(1 + LambertW(-LambertW(-exp(-1 + exp(-1) - exp(-1 + exp(-1)))))) * sqrt(1+LambertW(-exp(-1+exp(-1)-exp(-1+exp(-1))))) * sqrt(LambertW(-LambertW(-exp(-1+exp(-1)-exp(-1+exp(-1))))))). - Vaclav Kotesovec, Jan 29 2026

From Seiichi Manyama, Jun 01 2026: (Start)

E.g.f.: exp(B(x)), where B(x) is the e.g.f. of A396557.

a(0) = 1; a(n) = Sum_{i,j,k >= 0 and i+j+k=n-1} ((n-1)!/(i!*j!*k!)) * n^i * (-(n-i))^j * (k+2)^k. (End)

Maple

W := x -> LambertW(x): T := x -> -W(-x): LW := x -> W(-W(x))/(-W(x)):
ser := series(LW(T(x)), x, 24): seq(n!*coeff(ser, x, n), n=0..19);

Crossrefs

Cf. A340473, A097174, A177885, A207833, A227176.

Cf. A268653, A394239, A395376.

Cf. A000169, A000272, A396557.

Sequence in context: A330668 A001409 A260154 * A079489 A190526 A378258

Adjacent sequences: A340471 A340472 A340473 * A340475 A340476 A340477

Keyword

nonn

Author

Peter Luschny, Jan 09 2021

Status

approved