A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).

1, 4, 31, 301, 3274, 38158, 465919, 5883040, 76189177, 1006440238, 13508178448, 183689450959, 2525336086630, 35041483528522, 490125130328455, 6902993856515389, 97814486474787898, 1393470813699724726, 19946461692566594413, 286742046721454817358, 4138001844031453456120

Offset: 0

Author

Paul D. Hanna, Sep 07 2024

Keywords

• nonn

Comments

In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then

(C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).

(C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).

(C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).

Examples

			G.f. A(x) = 1 + 4*x + 31*x^2 + 301*x^3 + 3274*x^4 + 38158*x^5 + 465919*x^6 + 5883040*x^7 + 76189177*x^8 + 1006440238*x^9 + 13508178448*x^10 + ...
where A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
RELATED SERIES.
Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
B(x) = 1 + 4*x + 15*x^2 + 57*x^3 + 216*x^4 + 819*x^5 + 3105*x^6 + 11772*x^7 + ... + A125145(n)*x^n + ...
where B(x) = (1 + x)/(1 - 3*x - 3*x^2).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A125145, A215661, A375435, A375436, A375437.

Cf. A007863, A216314, A364374.

Programs

PARI

{a(n) = my(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + 3*x*(A+x*O(x^n))^2)); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n) = polcoef( (1/x)*serreverse( x*(1-3*x-3*x^2)/(1+x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n) = my(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2 * 3^j * A^j)*x^m/m))); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies:

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

(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 3^k * A(x)^k ).

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

(4) A(x) = Sum_{n>=0} A125145(n) * x^n * A(x)^n, where g.f. of A125145 = (1 + x)/(1 - 3*x - 3*x^2).

(5) x = (sqrt(21*A(x)^2 - 6*A(x) + 1) - (1 + 3*A(x)))/(6*A(x)^2).

a(n) = Sum_{k=0..n} 3^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024

a(n) ~ ((36 + (48266 - 714*sqrt(17))^(1/3) + (48266 + 714*sqrt(17))^(1/3))/7)^n / (sqrt(6*Pi*((20517 - 4861*sqrt(17))^(1/3) + (20517 + 4861*sqrt(17))^(1/3) - 42)) * n^(3/2)). - Vaclav Kotesovec, Sep 14 2024