A301928 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(2*n^2) / (1 + x*A(x)^(2*n))^n.

1, 1, 2, 9, 52, 338, 2429, 18656, 150974, 1276421, 11203817, 101708780, 952618462, 9192058947, 91313739683, 933740464780, 9830917830954, 106627370291103, 1192178247304598, 13750270811493561, 163695850973969924, 2012316205222767445, 25547555377097621270, 334912223771236381521, 4531556478213917470110

Offset

0,3

Comments

Note that if G(x) = Sum_{n>=0} x^n*G(x)^(2*n^2)/(1 + x*G(x)^(2*n))^(n+1), then G(x) = 1.

Note that if C(x) = Sum_{n>=0} x^n*C(x)^(2*n) / (1 + x*C(x)^2)^n, then C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).

Links

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 52*x^4 + 338*x^5 + 2429*x^6 + 18656*x^7 + 150974*x^8 + 1276421*x^9 + 11203817*x^10 + 101708780*x^11 + ...
such that
A(x) =  1 + x*A(x)^2/(1 + x*A(x)^2) + x^2*A(x)^8/(1 + x*A(x)^4)^2 + x^3*A(x)^18/(1 + x*A(x)^6)^3 + x^4*A(x)^32/(1 + x*A(x)^8)^4 + x^5*A(x)^50/(1 + x*A(x)^10)^5 + x^6*A(x)^72/(1 + x*A(x)^12)^6 +...

Prog

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

Crossrefs

Cf. A221585.

Sequence in context: A091319 A193465 A003584 * A069271 A305987 A355789

Adjacent sequences: A301925 A301926 A301927 * A301929 A301930 A301931

Keyword

nonn

Author

Paul D. Hanna, May 07 2018

Status

approved