A186998 G.f. satisfies: A(x) = Sum_{n>=0} x^n*(1 + x*A(x)^n)^n.

1, 1, 2, 4, 11, 35, 123, 462, 1829, 7558, 32380, 143102, 649999, 3026171, 14411412, 70095713, 347817785, 1759198500, 9063638685, 47545501777, 253854457415, 1379172691108, 7623064091313, 42860238300826, 245098499411379, 1425403070154478, 8429327482836740, 50681175605982771

Offset

0,3

Comments

Compare g.f. to a g.f. C(x) of the Catalan sequence:

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. satisfies: A(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2) / (1 - x*A(x)^n)^(n+1). - Paul D. Hanna, Sep 24 2014

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 35*x^5 + 123*x^6 +...
such that
A(x) = 1 + x*(1+x*A(x)) + x^2*(1+x*A(x)^2)^2 + x^3*(1+x*A(x)^3)^3 + x^4*(1+x*A(x)^4)^4 + x^5*(1+x*A(x)^5)^5 + x^6*(1+x*A(x)^6)^6 +...
The g.f. satisfies the series identity:
A(x) = 1/(1-x) + x^2*A(x)/(1-x*A(x))^2 + x^4*A(x)^4/(1-x*A(x)^2)^3 + x^6*A(x)^9/(1-x*A(x)^3)^4 + x^8*A(x)^16/(1-x*A(x)^4)^5 + x^10*A(x)^25/(1-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*(1+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 - x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 24 2014

Crossrefs

Cf. A186999, A203000, A247330, A186999.

Cf. A300041, A300042, A300043.

Sequence in context: A285002 A340338 A107378 * A243788 A245465 A219861

Adjacent sequences: A186995 A186996 A186997 * A186999 A187000 A187001

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2011

Status

approved