A300049 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (2 + 3*x*A(x)^n)^n.

1, 2, 7, 26, 130, 764, 4969, 34586, 253771, 1944728, 15462850, 126936170, 1071860068, 9284365430, 82326763672, 746181872468, 6905102562337, 65185100425700, 627340932873019, 6152204560716188, 61457807651698609, 625212555146338418, 6475800662211653437, 68281122104004473972, 732794914168656994990, 8003359250151679857176

Offset

0,2

Comments

Compare to the identity: if G(x) = Sum_{n>=0} x^n * (c + b*x*G(x)^n)^n,

then G(x) = Sum_{n>=0} b^n * x^(2*n) * G(x)^(n^2) / (1 - c*x*G(x)^n)^(n+1).

Links

Table of n, a(n) for n=0..25.

Formula

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} x^n * (2 + 3*x*A(x)^n)^n.

(2) A(x) = Sum_{n>=0} 3^n * x^(2*n) * A(x)^(n^2) / (1 - 2*x*A(x)^n)^(n+1).

Example

G.f.: A(x) = 1 + 2*x + 7*x^2 + 26*x^3 + 130*x^4 + 764*x^5 + 4969*x^6 + 34586*x^7 + 253771*x^8 + 1944728*x^9 + 15462850*x^10 + ...
such that
A(x) = 1 + x*(2 + 3*x*A(x)) + x^2*(2 + 3*x*A(x)^2)^2 + x^3*(2 + 3*x*A(x)^3)^3 + x^4*(2 + 3*x*A(x)^4)^4 + x^5*(2 + 3*x*A(x)^5)^5 + x^6*(2 + 3*x*A(x)^6)^6 + ...
Also, the g.f. satisfies the identity:
A(x) = 1/(1 - 2*x) + 3*x^2*A(x)/(1 - 2*x*A(x))^2 + 3^2*x^4*A(x)^4/(1 - 2*x*A(x)^2)^3 + 3^3*x^6*A(x)^9/(1 - 2*x*A(x)^3)^4 + 3^4*x^8*A(x)^16/(1 - 2*x*A(x)^4)^5 + 3^5*x^10*A(x)^25/(1 - 2*x*A(x)^5)^6 + ...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*(2 + 3*x*(A+x*O(x^n))^m)^m)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, 3^k*x^(2*k)*A^(k^2)/(1 - 2*x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A186998, A203014.

Sequence in context: A036757 A358493 A341900 * A209005 A361242 A068870

Adjacent sequences: A300046 A300047 A300048 * A300050 A300051 A300052

Keyword

nonn

Author

Paul D. Hanna, Feb 25 2018

Status

approved