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

Author

Paul D. Hanna, Dec 20 2023

Keywords

• nonn

Examples

			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) - ...
		

Crossrefs

Cf. A153231.

Programs

Mathematica

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

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), ", "))

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