A361768 Expansion of o.g.f. A(x) = 1/F(oo,x) where F(oo,x) is the limit of the process F(n,x) = (F(n-1,x)^n - n^2*x^n)^(1/n) for n > 0, starting with F(0,x) = 1.

1, 1, 3, 10, 35, 130, 499, 1966, 7893, 32168, 132690, 552784, 2322094, 9823572, 41811597, 178903031, 769044018, 3319438968, 14380154747, 62500478960, 272448124262, 1190815525727, 5217483053052, 22910925764270, 100811396881651, 444418225515884, 1962579128519888

Offset

0,3

Comments

a(n+1)/a(n) tends to 4.6131... - Vaclav Kotesovec, Mar 30 2023

Links

Paul D. Hanna, Table of n, a(n) for n = 0..600

Example

G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 35*x^4 + 130*x^5 + 499*x^6 + 1966*x^7 + 7893*x^8 + 32168*x^9 + 132690*x^10 + 552784*x^11 + ...
where A(x) = 1/F(oo,x) where F(oo,x) is the limit of the following process.
Start with F(0,x) = 1, and continue F(n,x) = (F(n-1,x)^n - n^2*x^n)^(1/n) for n > 0, like so:
F(0,x) = 1
F(1,x) = (F(0,x)^1 - 1^2*x^1)^(1/1)
F(2,x) = (F(1,x)^2 - 2^2*x^2)^(1/2)
F(3,x) = (F(2,x)^3 - 3^2*x^3)^(1/3)
F(4,x) = (F(3,x)^4 - 4^2*x^4)^(1/4)
F(5,x) = (F(4,x)^5 - 5^2*x^5)^(1/5)
...
then A(x) is the limit of 1/F(n,x) as n approaches infinity:
1/F(1,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...
1/F(2,x) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 141*x^6 + ...
1/F(3,x) = 1 + x + 3*x^2 + 10*x^3 + 31*x^4 + 105*x^5 + 363*x^6 + ...
1/F(4,x) = 1 + x + 3*x^2 + 10*x^3 + 35*x^4 + 125*x^5 + 463*x^6 + ...
1/F(5,x) = 1 + x + 3*x^2 + 10*x^3 + 35*x^4 + 130*x^5 + 493*x^6 + ...
...

Prog

(PARI) {a(n) = my(A=1, F); F=1; for(m=1, n, F = (F^m - m^2*x^m +x*O(x^(n+1)))^(1/m)); A = 1/F; H=A; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A361767, A361769.

Sequence in context: A303730 A149037 A228769 * A296164 A151046 A221130

Adjacent sequences: A361765 A361766 A361767 * A361769 A361770 A361771

Keyword

nonn

Author

Paul D. Hanna, Mar 29 2023

Status

approved