A369551 -id:A369551 - cp's OEIS frontend

A369091 Expansion of e.g.f. A(x) satisfying A(x) = x + A( x^2*exp(x) ), with A(0) = 0.

1, 2, 6, 36, 260, 2190, 21882, 268856, 3907080, 63977850, 1152946190, 22581979332, 477140664156, 10828556474918, 263163922847490, 6836792356168560, 189694001088036752, 5614994984290505586, 176964200467784915094, 5921022573291003915260, 209568707084236321665060

Offset: 1

Paul D. Hanna, Jan 26 2024

nonn

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

			E.g.f.: A(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! + 3907080*x^9/9! + 63977850*x^10/10! + ...
which equals the sum of all iterations of the function x^2*exp(x).
RELATED SERIES.
x*exp(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! + ... + A369090(n)*x^n/n! + ...
Let R(x) be the series reversion of A(x),
R(x) = x - 2*x^2/2! + 6*x^3/3! - 36*x^4/4! + 340*x^5/5! - 3870*x^6/6! + 52038*x^7/7! - 850472*x^8/8! + 16378920*x^9/9! + ...
then R(x) and e.g.f. A(x) satisfy:
(1) R( A(x) ) = x,
(2) R( A(x) - 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 the sum
A(x) = F(0) + F(1) + F(2) + F(3) + ... + F(n) + ...
equivalently,
A(x) = 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))) + ...

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

Cf. A369090, A369551 (a(n)/n), A030178.

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

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = x + A( x^2*exp(x) ).
(2) A(x) = Sum_{n>=0} F(n), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
(3) A(x) = log(G(x)/x) where G(x) = G(x^2*exp(x))/x is the e.g.f. of A369090.

A369550 Expansion of e.g.f. A(x) satisfying A(x) = exp(x) * A(x^2*exp(x)).

Original entry on oeis.org

1, 1, 3, 13, 85, 701, 6901, 79045, 1049385, 15924025, 271248121, 5108389001, 105158055949, 2346022349269, 56348945801877, 1449434215375021, 39758549273200081, 1159092552400164977, 35813081725133941297, 1169791166246561367697, 40297553373717279300981, 1460613225168596836153741

Offset: 0

Paul D. Hanna, Jan 29 2024

nonn

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

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! + 6901*x^6/6! + 79045*x^7/7! + 1049385*x^8/8! + 15924025*x^9/9! + ...
RELATED SERIES.
The expansion of A(x^2*exp(x)) begins
exp(-x) * A(x) = A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 380*x^5/5! + 3750*x^6/6! +  + 42882*x^7/7! + 576296*x^8/8! + ...
The logarithm of e.g.f. A(x) equals L(x) where L(x) = x + L(x^2*exp(x)),
L(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! + ...

Cf. A369090, A369091, A369551.

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

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = exp(x) * A(x^2*exp(x)).
(2) A(x) = exp( Sum_{n>=0} F(n) ), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
(3) A(x) = exp(L(x)) where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
(4) A(x) = G(x)/x where G(x) = G(x^2*exp(x))/x is the e.g.f. of A369090.
a(n) = A369090(n+1)/(n+1) for n >= 0.

cp's OEIS Frontend

A369091 Expansion of e.g.f. A(x) satisfying A(x) = x + A( x^2*exp(x) ), with A(0) = 0.

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