OFFSET
1,2
COMMENTS
E.g.f. equals the logarithm of G(x)/x, where G(x) equals the e.g.f. of A277180.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 144*x^4/4! + 1660*x^5/5! + 27480*x^6/6! + 548394*x^7/7! + 12402992*x^8/8! + 316789848*x^9/9! + 9158652720*x^10/10! + 296955697390*x^11/11! + 10666960742328*x^12/12! + 420121365404052*x^13/13! + 17973670280757464*x^14/14! + 828915057583647090*x^15/15! + 40974375613614916320*x^16/16! +...
such that x*exp(A(x)) equals the infinite composition of functions:
x*exp(A(x)) = ... o x*exp(x^4) o x*exp(x^3) o x*exp(x^2) o x*exp(x),
which expands to begin:
x*exp(A(x)) = x + 2*x^2/2! + 9*x^3/3! + 100*x^4/4! + 1205*x^5/5! + 18006*x^6/6! + 350077*x^7/7! + 8088536*x^8/8! + 211371561*x^9/9! + 6176234890*x^10/10! + 200898827921*x^11/11! + 7219180413732*x^12/12! +...+ A277180(n)*x^n/n! +...
A related series expansion begins
exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 241*x^4/4! + 3001*x^5/5! + 50011*x^6/6! + 1011067*x^7/7! + 23485729*x^8/8! + 617623489*x^9/9! + 18263529811*x^10/10! + 601598367811*x^11/11! + 21859800985969*x^12/12! +...
GENERATING METHOD.
Once can generate the e.g.f. by the following process.
Start with L_1 = x, then continue
L_2 = L_1 + x^2*exp(2*L_1)
L_3 = L_2 + x^3*exp(3*L_2)
L_4 = L_3 + x^4*exp(4*L_3)
...
L_{n+1} = L_{n} + x^(n+1)*exp( (n+1)*L_{n} )
...
which tends to e.g.f. A(x) as a limit.
Explicitly, the initial functions are:
L_1 = x
L_2 = x + x^2*exp(2*x)
L_3 = x + x^2*exp(2*x) + x^3*exp(3*x + 3*x^2*exp(2*x) )
L_4 = x + x^2*exp(2*x) + x^3*exp(3*x + 3*x^2*exp(2*x) ) + x^4*exp(4*x + 4*x^2*exp(2*x) + 4*x^3*exp(3*x + 3*x^2*exp(2*x) ) )
L_5 = x + x^2*exp(2*x) + x^3*exp( 3*x + 3*x^2*exp(2*x) ) + x^4*exp( 4*x + 4*x^2*exp(2*x) + 4*x^3*exp( 3*x + 3*x^2*exp(2*x) ) ) + x^5*exp( 5*x + 5*x^2*exp(2*x) + 5*x^3*exp( 3*x + 3*x^2*exp(2*x) ) + 5*x^4*exp( 4*x + 4*x^2*exp(2*x) + 4*x^3*exp( 3*x + 3*x^2*exp(2*x) ) ) )
...
The derivative of these series may be computed like so
L_1' = 1
L_2' = L_1' + 2*x^1*exp( 2*L_1 ) * (1 + x*L_1')
L_3' = L_2' + 3*x^2*exp( 3*L_2 ) * (1 + x*L_2')
L_4' = L_3' + 4*x^3*exp( 4*L_3 ) * (1 + x*L_3')
...
PROG
(PARI) {a(n) = my(A=x +x^2*O(x^n)); if(n<=0, 0, for(i=1, n, A = A*exp(A^i)); n!*polcoeff(log(A/x), n))}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=x +x^2*O(x^n)); if(n<=0, 0, for(i=1, n, A = subst(A, x, x*exp(x^(n-i+1) +x*O(x^n))))); n!*polcoeff(log(A/x), n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 06 2016
STATUS
approved