A300043 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * (1 + x*A(x)^n)^n.

1, 1, 3, 10, 40, 176, 830, 4115, 21198, 112559, 612632, 3404041, 19251182, 110558737, 643625347, 3792942385, 22602071988, 136073693419, 827100093078, 5073042015385, 31385390615698, 195795705705357, 1231410381962373, 7806727084866309, 49885661271015892, 321311108707030967, 2086162636742801262, 13655153319525415679, 90125387322138673911

Offset

0,3

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 * A(x)^n * (1 + x*A(x)^n)^n.

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

(3) A(x) = (1/x) * Series_Reversion( x/G(x) ), where G(x) = A(x/G(x)) is the g.f. of A300041.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 40*x^4 + 176*x^5 + 830*x^6 + 4115*x^7 + 21198*x^8 + 112559*x^9 + 612632*x^10 + 3404041*x^11 + 19251182*x^12 + ...
such that
A(x) = 1 + x*A(x)*(1+x*A(x)) + x^2*A(x)^2*(1+x*A(x)^2)^2 + x^3*A(x)^3*(1+x*A(x)^3)^3 + x^4*A(x)^4*(1+x*A(x)^4)^4 + x^5*A(x)^5*(1+x*A(x)^5)^5 + x^6*A(x)^6*(1+x*A(x)^6)^6 + ...
The g.f. also satisfies the series identity:
A(x) = 1/(1-x*A(x)) + x^2*A(x)^2/(1-x*A(x)^2)^2 + x^4*A(x)^6/(1-x*A(x)^3)^3 + x^6*A(x)^12/(1-x*A(x)^4)^4 + x^8*A(x)^20/(1-x*A(x)^5)^5 + x^10*A(x)^30/(1-x*A(x)^6)^6 + ...

Prog

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

Crossrefs

Cf. A300041, A186998, A300042.

Sequence in context: A151075 A151076 A151077 * A258973 A217885 A216367

Adjacent sequences: A300040 A300041 A300042 * A300044 A300045 A300046

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2018

Status

approved