A352383 G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x)^3)^2 / (1 + x*A(x)^3).

1, 3, 28, 350, 5020, 78023, 1278340, 21740636, 380161308, 6792111260, 123448657904, 2275311657814, 42427160829508, 798933055335618, 15171376583787800, 290199619787772728, 5586346847185229596, 108141141737193646020

Offset

0,2

Comments

Equals a bisection of A351771.

Self-convolution cube equals A352384.

Links

Table of n, a(n) for n=0..17.

Formula

G.f. A(x) satisfies:

(1) A(x) = (1 + 2*x*A(x)^3)^2 / (1 + x*A(x)^3).

(2) 0 = 4*A(x)^6*x^2 + (4 - A(x))*A(x)^3*x + (1 - A(x)).

(3) A(x) = 4 + 8*x*A(x)^3 - sqrt( A(x)^2 + 8*A(x) ).

(4) x = ( A(x) - 4 + sqrt( A(x)^2 + 8*A(x) ) ) / (8*A(x)^3).

(5) A( x*(1+x)^3/(1+2*x)^6 ) = (1+2*x)^2/(1+x).

(6) A(x)^3 = (1/x) * Series_Reversion( x*(1+x)^3/(1+2*x)^6 ).

a(n) ~ sqrt((221 - 29*sqrt(13))/78) * 2^(3*n) * (587 - 143*sqrt(13))^n / (sqrt(Pi) * n^(3/2) * 3^(3*n+1)). - Vaclav Kotesovec, Mar 15 2022

Example

G.f.: A(x) = 1 + 3*x + 28*x^2 + 350*x^3 + 5020*x^4 + 78023*x^5 + 1278340*x^6 + 21740636*x^7 + 380161308*x^8 + 6792111260*x^9 + ...
where A(x) satisfies
A(x) = (1 + 2*x*A(x)^3)^2 / (1 + x*A(x)^3).
Related series.
The cube of the g.f. A(x) equals the g.f. of A352384:
A(x)^3 = 1 + 9*x + 111*x^2 + 1581*x^3 + 24468*x^4 + 399735*x^5 + 6784186*x^6 + 118444293*x^7 + ... + A352384(n)*x^n + ...
and equals (1/x) * Series_Reversion(  x*(1+x)^3/(1+2*x)^6 ).

Mathematica

CoefficientList[(InverseSeries[Series[x*(1 + x)^3/(1 + 2*x)^6, {x, 0, 20}], x]/x)^(1/3), x] (* Vaclav Kotesovec, Mar 15 2022 *)

Prog

(PARI) /* Using Series Reversion */
{a(n) = my(A = ((1/x)*serreverse( x*(1+x)^3/(1+2*x +x^2*O(x^n))^6 ))^(1/3)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A352384, A351771, A188110.

Sequence in context: A072343 A292845 A212032 * A151423 A161605 A048954

Adjacent sequences: A352380 A352381 A352382 * A352384 A352385 A352386

Keyword

nonn

Author

Paul D. Hanna, Mar 14 2022

Status

approved