A300042 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + x*A(x)^(n+1))^n.

1, 1, 2, 5, 16, 58, 228, 949, 4130, 18633, 86622, 413106, 2014489, 10020342, 50748198, 261324021, 1366804389, 7255452421, 39066835030, 213287955245, 1180397594359, 6621150605830, 37641036925947, 216882646869800, 1266660408267898, 7499333882769716, 45017192824063767, 274030099624436499, 1691811333997049888, 10595032219552021063, 67313254111562228356

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..100

Formula

G.f. satisfies:

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

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 58*x^5 + 228*x^6 + 949*x^7 + 4130*x^8 + 18633*x^9 + 86622*x^10 + 413106*x^11 + 2014489*x^12 + ...
such that
A(x) = 1 + x*(1+x*A(x)^2) + x^2*(1+x*A(x)^3)^2 + x^3*(1+x*A(x)^4)^3 + x^4*(1+x*A(x)^5)^4 + x^5*(1+x*A(x)^6)^5 + x^6*(1+x*A(x)^7)^6 + ...
The g.f. also satisfies the series identity:
A(x) = 1/(1-x) + x^2*A(x)^2/(1-x*A(x))^2 + x^4*A(x)^6/(1-x*A(x)^2)^3 + x^6*A(x)^12/(1-x*A(x)^3)^4 + x^8*A(x)^20/(1-x*A(x)^4)^5 + x^10*A(x)^30/(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+1))^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*(k+1))/(1 - x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A300041, A186998, A300043.

Sequence in context: A149979 A262441 A328296 * A019448 A243326 A185143

Adjacent sequences: A300039 A300040 A300041 * A300043 A300044 A300045

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2018

Status

approved