A361049 G.f. satisfies: A(x) = (1/x)*Series_Reversion( x/(1 + x*A(x)^2 + x^2*A(x)*A'(x)) ).

1, 1, 4, 28, 269, 3201, 44737, 711691, 12630023, 246594988, 5244025502, 120540052304, 2976918491501, 78601791684495, 2209667973082374, 65901745111752843, 2078619947109354811, 69141776287740239348, 2419303138068147399700, 88842295496847889690405

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

Given g.f. A(x) = Sum_{n>=0} a(n)*x^n, let G(x) be the g.f. of A361048, then the following formulas hold.

(1) A(x) = (1/x)*Series_Reversion( x/(1 + x*A(x)^2 + x^2*A(x)*A'(x)) ).

(2) A(x) = G(x*A(x)).

(4) A(x) = (1/x)*Series_Reversion(x/G(x)).

(3) G(x) = A(x/G(x)).

(5) G(x) = 1 + d/dx (x^2 * A(x)^2)/2.

a(n) ~ c * n! * n^(3*LambertW(1) - 2 + 3/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.13835030685615842626... - Vaclav Kotesovec, Mar 13 2023

Example

G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 269*x^4 + 3201*x^5 + 44737*x^6 + 711691*x^7 + 12630023*x^8 + 246594988*x^9 + ...
such that A(x) = G(x*A(x)) where G(x) is given by
G(x) = 1 + d/dx (x^2 * A(x)^2)/2, which begins
G(x) = 1 + x + 3*x^2 + 18*x^3 + 160*x^4 + 1830*x^5 + 25074*x^6 + 395248*x^7 + 6990876*x^8 + ... + A361048(n)*x^n + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n,
A = (1/x)*serreverse( x/(1 + x*A^2 + x^2*A*A') +x^2*O(x^n) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A361048, A132070, A113882.

Sequence in context: A316144 A138272 A367470 * A372747 A245060 A191686

Adjacent sequences: A361046 A361047 A361048 * A361050 A361051 A361052

Keyword

nonn

Author

Paul D. Hanna, Mar 13 2023

Status

approved