A375435 Expansion of g.f. A(x) satisfying A(x) = (1 + 3*x*A(x)) * (1 + x*A(x)^2).

1, 4, 23, 167, 1370, 12066, 111399, 1063896, 10423145, 104172842, 1057938416, 10886055709, 113252336950, 1189231665334, 12588038915535, 134172815937543, 1438842536532522, 15513036330871914, 168057711839246901, 1828443841807079994, 19970180509170366264, 218877585875869278396

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 + 23*x^2 + 167*x^3 + 1370*x^4 + 12066*x^5 + 111399*x^6 + 1063896*x^7 + 10423145*x^8 + 104172842*x^9 + 1057938416*x^10 + ...
where A(x) = (1 + 3*x*A(x)) * (1 + 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 + 7*x^2 + 19*x^3 + 40*x^4 + 97*x^5 + 217*x^6 + 508*x^7 + 1159*x^8 + ... + A006130(n+1)*x^n + ...
where B(x) = (1 + 3*x)/(1 - x - 3*x^2).
		

Links

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

Crossrefs

Cf. A006130, A216314, A215661, A375434, A375436, A375437.

Programs

PARI

{a(n) = my(A=1+x); for(i=1, n, A=(1 + 3*x*A)*(1 + 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 - x - 3*x^2)/(1+3*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^(m-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 the following formulas.

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

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

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

(4) A(x) = Sum_{n>=0} A006130(n+1) * x^n * A(x)^n, where g.f. of A006130 = 1/(1 - x - 3*x^2).

(5) x = (sqrt(13*A(x)^2 - 6*A(x) + 9) - (3 + A(x)))/(6*A(x)^2).

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