A180610 G.f. satisfies: L(x) = L(x*exp(x))/(1+x) = Sum_{n>=1} a(n)*x^n/n!^2.

1, -2, 15, -240, 6420, -249120, 12729360, -799249920, 59539354560, -5472506188800, 708462047462400, -129547455170918400, 24363744478955481600, -2357904494544779980800, -506327043021030975744000

Offset

1,2

Links

Table of n, a(n) for n=1..15.

Formula

G.f. L(x) satisfies: L(x) = (1 + W(x))*L(W(x)) where W(x) = Sum_{n>=1} (-1)^(n-1)*n^n*x^n/n! is the LambertW function.

Let E_n(x) = E_{n-1}(x*exp(x)) denote the n-th iteration of x*exp(x), then

. L(E_2(x)) = L(x)*(1+x)*(1+x*exp(x));

. L(E_3(x)) = L(x)*(1+x)*(1+x*exp(x))*(1+x*exp(x*exp(x)));

. L(E_n(x)) = L(x)*x*E_n'(x)/E_n(x) = L(x)*Product_{k=0..n-1}(1+E_k(x)).

G.f. L(x) forms column 0 in the matrix log of the Riordan array (exp(x), x*exp(x)).

Example

G.f: L(x) = x - 2*x^2/2!^2 + 15*x^3/3!^2 - 240*x^4/4!^2 + 6420*x^5/5!^2 - 249120*x^6/6!^2 + 12729360*x^7/7!^2 -+...
The Riordan array (exp(x), x*exp(x)) begins:
1;
1, 1;
1/2!, 2, 1;
1/3!, 4/2!, 3, 1;
1/4!, 8/3!, 9/2!, 4, 1;
1/5!, 16/4!, 27/3!, 16/2!, 5, 1;
1/6!, 32/5!, 81/4!, 64/3!, 25/2!, 6, 1; ...
The matrix log of the above array begins:
0;
1, 0;
-2/2!^2, 2, 0;
15/3!^2, -4/2!^2, 3, 0;
-240/4!^2, 30/3!^2, -6/2!^2, 4, 0;
6420/5!^2, -480/4!^2, 45/3!^2, -8/2!^2, 5, 0;
-249120/6!^2, 12840/5!^2, -720/4!^2, 60/3!^2, -10/2!^2, 6, 0; ...
in which the g.f. of column k equals (k+1)*L(x) for k>=0 where L(x) is the g.f. of this sequence.

Prog

(PARI) {a(n)=local(M=matrix(n+1, n+1, r, c, if(r>=c, polcoeff(exp(c*x+x*O(x^n)), r-c))), L=sum(n=1, #M, -(M^0-M)^n/n)); n!^2*L[n+1, 1]}

Crossrefs

Sequence in context: A192561 A356587 A227098 * A156750 A292798 A379886

Adjacent sequences: A180607 A180608 A180609 * A180611 A180612 A180613

Keyword

eigen,sign

Author

Paul D. Hanna, Sep 11 2010

Extensions

Formula corrected by Paul D. Hanna, Sep 19 2010

Status

approved