A300041 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + x*A(x)^(n-1))^n.

1, 1, 2, 3, 7, 20, 63, 215, 783, 2998, 11977, 49656, 212738, 938836, 4257792, 19808597, 94405713, 460412410, 2295740045, 11695447378, 60837384509, 322968172763, 1748975296265, 9657311996480, 54350006350292, 311630231535041, 1819713622889812, 10817233370816701, 65434087495967354, 402615569685977397, 2518832660928798529, 16016013141937173805

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).

(3) A(x) = x/Series_Reversion( x*G(x) ), where G(x) = A(x*G(x)) is the g.f. of A300043.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 20*x^5 + 63*x^6 + 215*x^7 + 783*x^8 + 2998*x^9 + 11977*x^10 + 49656*x^11 + 212738*x^12 + ...
such that
A(x) = 1 + x*(1+x) + x^2*(1+x*A(x))^2 + x^3*(1+x*A(x)^2)^3 + x^4*(1+x*A(x)^3)^4 + x^5*(1+x*A(x)^4)^5 + x^6*(1+x*A(x)^5)^6 + ...
The g.f. also satisfies the series identity:
A(x) = 1/(1-x) + x^2/(1-x*A(x))^2 + x^4*A(x)^2/(1-x*A(x)^2)^3 + x^6*A(x)^6/(1-x*A(x)^3)^4 + x^8*A(x)^12/(1-x*A(x)^4)^5 + x^10*A(x)^20/(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. A186998, A300042, A300043.

Sequence in context: A293683 A006073 A052402 * A222939 A222867 A222776

Adjacent sequences: A300038 A300039 A300040 * A300042 A300043 A300044

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2018

Status

approved