OFFSET
0,4
COMMENTS
Limit a(n)/(n-1)! = 1.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..449
FORMULA
a(n) ~ (n-1)! * (1 + 1/n^3 + 11/n^4 + 98/n^5 + 879/n^6 + 8491/n^7 + 90523/n^8 + 1072357/n^9 + 14094094/n^10 + ...). - Vaclav Kotesovec, Dec 27 2020
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 121*x^6 + 726*x^7 + 5079*x^8 + 40560*x^9 + 364476*x^10 + 3640206*x^11 + 40006088*x^12 +...
Illustration of definition.
Start with R1 = A(x) - x, and proceed as follows:
R2 = (R1 - x^2)^(1/2) = 1 + x^3 + 3*x^4 + 12*x^5 + 60*x^6 + 360*x^7 + 2523*x^8 +...
R3 = (R2 - x^3)^(1/3) = 1 + x^4 + 4*x^5 + 20*x^6 + 120*x^7 + 840*x^8 +...
R4 = (R3 - x^4)^(1/4) = 1 + x^5 + 5*x^6 + 30*x^7 + 210*x^8 + 1680*x^9 +...
R5 = (R4 - x^5)^(1/5) = 1 + x^6 + 6*x^7 + 42*x^8 + 336*x^9 + 3024*x^10 +...
R6 = (R5 - x^6)^(1/6) = 1 + x^7 + 7*x^8 + 56*x^9 + 504*x^10 + 5040*x^11 +...
R7 = (R6 - x^7)^(1/7) = 1 + x^8 + 8*x^9 + 72*x^10 + 720*x^11 + 7920*x^12 +...
R8 = (R7 - x^8)^(1/8) = 1 + x^9 + 9*x^10 + 90*x^11 + 990*x^12 + 11880*x^13 +...
R9 = (R8 - x^9)^(1/9) = 1 + x^10 + 10*x^11 + 110*x^12 + 1320*x^13 + 17160*x^14 +...
etc., to approach the value 1 as a limit.
Generating Method.
The g.f. may be attained as a limit of the following process.
Start with 1, add x^n and raise that result to the (n-1) power, add x^(n-1) and raise that result to the (n-2) power, and continue until you reach x^1 to approximate the g.f. A(x).
PROG
(PARI) {a(n) = my(A=1); for(k=1, n, A = A^(n+1-k) + x^(n+1-k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 16 2016
STATUS
approved