A369090 Expansion of e.g.f. A(x) satisfying A(x) = A( x^2*exp(x) ) / x, with A(0) = 0.

1, 2, 9, 52, 425, 4206, 48307, 632360, 9444465, 159240250, 2983729331, 61300668012, 1367054727337, 32844312889766, 845234187028155, 23190947446000336, 675895337644401377, 20863665943202969586, 680448552777544884643, 23395823324931227353940, 846248620848062865320601

Offset

1,2

Comments

Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.

(1) A(x) = A(x^2*exp(x)) / x.

(2) R(x*A(x)) = x^2*exp(x), where R(A(x)) = x.

(3) A(x) = x * exp( Sum_{n>=0} F(n) ), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.

(4) A(x) = x * exp(L(x)), where L(x) = x + L(x^2*exp(x)) is the e.g.f. of A369091.

Example

E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! + 4206*x^6/6! + 48307*x^7/7! + 632360*x^8/8! + 9444465*x^9/9! + 159240250*x^10/10! + ...
RELATED SERIES.
The expansion of the logarithm of A(x)/x starts
log(A(x)/x) = x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + ... + A369091(n)*x^n/n! + ...
and equals the sum of all iterations of the function x^2*exp(x).
Let R(x) be the series reversion of A(x),
R(x) = x - 2*x^2/2! + 3*x^3/3! + 8*x^4/4! - 155*x^5/5! + 1464*x^6/6! - 7931*x^7/7! - 65360*x^8/8! + 2742345*x^9/9! + ...
then R(x) and e.g.f. A(x) satisfy:
(1) R( A(x) ) = x,
(2) R( x*A(x) ) = x^2 * exp(x).
GENERATING METHOD.
Let F(n) equal the n-th iteration of x^2*exp(x), so that
F(0) = x,
F(1) = x^2 * exp(x),
F(2) = x^4 * exp(2*x) * exp(x^2*exp(x)),
F(3) = x^8 * exp(4*x) * exp(2*x^2*exp(x)) * exp(F(2)),
F(4) = x^16 * exp(8*x) * exp(4*x^2*exp(x)) * exp(2*F(2)) * exp(F(3)),
F(5) = x^32 * exp(16*x) * exp(8*x^2*exp(x)) * exp(4*F(2)) * exp(2*F(3)) * exp(F(4)),
...
F(n+1) = F(n)^2 * exp(F(n))
...
Then the e.g.f. A(x) equals
A(x) = x * exp(F(0) + F(1) + F(2) + F(3) + ... + F(n) + ...).
equivalently,
A(x) = x * exp(x + x^2*exp(x) + x^4*exp(2*x)*exp(x^2*exp(x)) + x^8*exp(4*x)*exp(2*x^2*exp(x)) * exp(x^4*exp(2*x)*exp(x^2*exp(x))) + ...).

Prog

(PARI) {a(n) = my(A=x); for(i=0, #binary(n),
A = subst(A, x, x^2*exp(x +x^2*O(x^n)) )/x ); n! * polcoeff(H=A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A369091, A369550 (a(n)/n), A030178.

Cf. A367390.

Sequence in context: A330200 A143922 A305304 * A110322 A161631 A121678

Adjacent sequences: A369087 A369088 A369089 * A369091 A369092 A369093

Keyword

nonn

Author

Paul D. Hanna, Jan 26 2024

Status

approved