A013703 Series(W(exp(1)*(1+x)), x) = sum( a[ n ]/(2^(2*n)*n!), n=0..infinity), where W is the Lambert W function.

1, 2, -6, 38, -370, 4874, -81046, 1628710, -38393538, 1038795658, -31730277062, 1080038539942, -40538501660306, 1663428036271754, -74080097240364918, 3558651343664651174, -183423140013051563746, 10097324775041880827402, -591270189493633774009510

Offset

0,2

Comments

The n-th derivative of W(x) at x=exp(1) is exp(-n) * a(n) / 2^(2*n). - Paolo Bonzini, Jun 23 2016

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

Formula

E.g.f.: LambertW(exp(1)*(1+4*x)). - Vladeta Jovovic, Nov 19 2003

|a(n)| ~ 4^n * exp(n) * n^(n-1) / (1+exp(2))^(n-1/2). - Vaclav Kotesovec, Jul 09 2013

Example

LambertW(exp(1)*(1+4*x)) = 1 + 1/2*x - 3/16*x^2 + 19/192*x^3  - 185/3072*x^4 + ...

Maple

LambertW( exp(1)*(1+x) );
seq(n!*coeff(series(LambertW(exp(1)*(1+4*x)), x, n+1), x, n), n=0..20); # Vaclav Kotesovec, Jul 09 2013

Mathematica

max = 17; (Series[ ProductLog[E*(1 + 4*x)], {x, 0, max}] // CoefficientList[#, x] &)*Range[0, max]! (* Jean-François Alcover, Jun 20 2013 *)

Prog

(Maxima)
a(n):= if n<2 then n+1 else 2*(n-1)!*(sum((binomial(n+k-1, n-1)*sum(binomial(k, j)*2^(n-j-1)*sum(binomial(j, l)*(-1)^(l)*sum((l^(n+j-i-1)*binomial(l, i))/(n+j-i-1)!, i, 0, l), l, 1, j), j, 1, k)), k, 1, n-1)); \\ based on A042977, Paolo Bonzini, Jun 23 2016
(PARI) x='x+O('x^50); Vec(serlaplace(lambertw(exp(1)*(1+4*x)))) \\ G. C. Greubel, Nov 15 2017

Crossrefs

Twice row sums of A042977.

Sequence in context: A068184 A067106 A032111 * A002031 A184731 A005738

Adjacent sequences: A013700 A013701 A013702 * A013704 A013705 A013706

Keyword

sign

Author

Robert Corless (rmc(AT)pineapple.apmaths.uwo.ca)

Extensions

More terms from N. J. A. Sloane.

More terms from Vincenzo Librandi, Jul 25 2013

Status

approved