A322726 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (3 + x*A(x)^n)^n.

1, 3, 10, 36, 155, 825, 5227, 36930, 277933, 2181186, 17716100, 148600698, 1286066999, 11471959881, 105320978028, 993555579651, 9616293352921, 95370420276468, 968267603845781, 10056629892262203, 106798045980208919, 1159174549401667188, 12854190552592134081, 145574358625296958326, 1683040220764208777635

Offset

0,2

Links

Table of n, a(n) for n=0..24.

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + 3*x + 10*x^2 + 36*x^3 + 155*x^4 + 825*x^5 + 5227*x^6 + 36930*x^7 + 277933*x^8 + 2181186*x^9 + 17716100*x^10 + ...
such that
A(x) = 1 + x*(3 + x*A(x)) + x^2*(3 + x*A(x)^2)^2 + x^3*(3 + x*A(x)^3)^3 + x^4*(3 + x*A(x)^4)^4 + x^5*(3 + x*A(x)^5)^5 + x^6*(3 + x*A(x)^6)^6 + ...
Also, the g.f. satisfies the identity:
A(x) = 1/(1 - 3*x) + x^2*A(x)/(1 - 3*x*A(x))^2 + x^4*A(x)^4/(1 - 3*x*A(x)^2)^3 + x^6*A(x)^9/(1 - 3*x*A(x)^3)^4 + x^8*A(x)^16/(1 - 3*x*A(x)^4)^5 + x^10*A(x)^25/(1 - 3*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*(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, x^(2*k)*A^(k^2)/(1 - 3*x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A186998, A203014, A300049, A322725.

Sequence in context: A353262 A165792 A010373 * A104603 A337341 A080625

Adjacent sequences: A322723 A322724 A322725 * A322727 A322728 A322729

Keyword

nonn

Author

Paul D. Hanna, Jan 30 2019

Status

approved