A367389 Expansion of g.f. A(x) = B(x^2)/(1 - 2*x*B(x^2)) where B(x) = 1 + 2*x*B(x)^3 is the g.f. of A153231.

1, 2, 6, 16, 52, 152, 512, 1568, 5392, 16992, 59232, 190336, 669952, 2183680, 7742464, 25512448, 90974464, 302368256, 1083175424, 3625435136, 13036688384, 43889186816, 158323564544, 535639556096, 1937483350016, 6582584115200, 23865932414976, 81381420826624, 295661476642816

Offset

0,2

Links

Table of n, a(n) for n=0..28.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x) = B(x^2)/(1 - 2*x*B(x^2)) where B(x) = 1 + 2*x*B(x)^3 is the g.f. of A153231.

(2) A(x) = 1/(1-2*x - Sum_{n>=1} 2^n * binomial(3*n-1,n)/(3*n-1) * x^(2*n) ).

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

(4) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n-1)^(n-2) for n > 1.

(5) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = -n*(n-2)^(n-2) for n > 1.

(6) [x^(n-1)] (1 + (n-2)*x*A(x))^n / A(x)^n = -n*(5*n-14)*(n-4)^(n-3) for n >= 1.

a(n) ~ (15*sqrt(3/2)/2 + 9 + (15*sqrt(3/2)/2 - 9)*(-1)^n) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2) * 2^(n/2)). - Vaclav Kotesovec, Dec 24 2023

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 52*x^4 + 152*x^5 + 512*x^6 + 1568*x^7 + 5392*x^8 + 16992*x^9 + 59232*x^10 + ...
where
1/A(x) = 1 - 2*x - 2*x^2 - 8*x^4 - 56*x^6 - 480*x^8 - 4576*x^10 - 46592*x^12 - ... - 2^n*binomial(3*n-1,n)/(3*n-1) * x^(2*n) - ...

Mathematica

CoefficientList[1/(-2*x + x/InverseSeries[Series[x - 2*x^3, {x, 0, 30}], x]), x] (* Vaclav Kotesovec, Dec 24 2023 *)

Prog

(PARI) {a(n) = my(A = 1/(-2*x + x/serreverse(x - 2*x^3 + O(x^(n+2))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A;
A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m-1)^(m-2) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A153231.

Sequence in context: A258797 A360232 A214983 * A214833 A263593 A003446

Adjacent sequences: A367386 A367387 A367388 * A367390 A367391 A367392

Keyword

nonn

Author

Paul D. Hanna, Dec 20 2023

Status

approved