OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. A(x) satisfies:
(1) Sum_{n>=0} (n+1 - A(x))^n * x^n/n! = A(x).
(2) Sum_{n>=0} (n+2 - A(x))^n * x^n/n! = A(x) * LambertW(-x)/(-x).
(3) Sum_{n>=0} (n+m - A(x))^n * x^n/n! = A(x) * ( LambertW(-x)/(-x) )^(m-1).
(4) Sum_{n>=0} (n+1 - 2*A(x))^n * x^n/n! = A(x)^2 * (1 + LambertW(-x)) / ( LambertW(-x)/(-x) ).
(5) Sum_{n>=0} (n+m - p*A(x))^n * x^n/n! = A(x)^p * (1 + LambertW(-x))^(p-1) * ( LambertW(-x)/(-x) )^(m-p).
(6) A(x) = ( LambertW(-x)/(-x) )^(1 - A(x)) / (1 + LambertW(-x)).
(7) A(x) = exp( LambertW(-x)*A(x) ) * ( LambertW(-x)/(-x) ) / (1 + LambertW(-x)).
E.g.f.: -LambertW(LambertW(-x)^2/(x + x*LambertW(-x))) / LambertW(-x). - Vaclav Kotesovec, Sep 01 2020
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 865*x^5/5! + 11976*x^6/6! + 197449*x^7/7! + 3798992*x^8/8! + 83513889*x^9/9! + 2063157760*x^10/10! + ...
where
A(x) = 1 + (2 - A(x))*x + (3 - A(x))^2*x^2/2! + (4 - A(x))^3*x^3/3! + (5 - A(x))^4*x^4/4! + (6 - A(x))^5*x^5/5! + (7 - A(x))^6*x^6/6! + ...
Also,
A(x) * LambertW(-x)/(-x) = 1 + (3 - A(x))*x + (4 - A(x))^2*x^2/2! + (5 - A(x))^3*x^3/3! + (6 - A(x))^4*x^4/4! + (7 - A(x))^5*x^5/5! + ...
where
LambertW(-x)/(-x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + ... + (n+1)^(n-1)*x^n/n! + ...
MATHEMATICA
nmax = 20; CoefficientList[Series[-LambertW[LambertW[-x]^2/(x + x*LambertW[-x])]/LambertW[-x], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 01 2020 *)
PROG
(PARI) {a(n) = my(A=1 +O(x)); for(i=1, n, A = sum(m=0, n, (m+1 - A)^m *x^m/m!) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 31 2020
STATUS
approved