A237645 G.f. satisfies: A(x) = G(x*A(x)) where G(x) = -1+x + A(x) + 1/A(x).

1, 1, 2, 7, 34, 201, 1357, 10109, 81397, 698948, 6341597, 60391832, 600661215, 6215862360, 66726103981, 741259084280, 8504902411004, 100618874020119, 1225724374602147, 15356200178917791, 197646961110310062, 2610956607315266757, 35370366025297098315

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. satisfies:

(1) A(x) = -1 + x*A(x) + A(x*A(x)) + 1/A(x*A(x)).

(2) A(x) = (1/x) * Series_Reversion( x / ( -1+x + A(x) + 1/A(x) ) ).

a(n) = [x^n] ( -1+x + A(x) + 1/A(x) )^(n+1) / (n+1).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 201*x^5 + 1357*x^6 +...
Let G(x) = -1+x + A(x) + 1/A(x):
G(x) = 1 + x + x^2 + 3*x^3 + 13*x^4 + 70*x^5 + 436*x^6 + 3024*x^7 + 22828*x^8 + 184795*x^9 + 1587809*x^10 +...
then A(x) = G(x*A(x)) and G(x) = A(x/G(x)).
Related expansions.
A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 72*x^4 + 470*x^5 + 3449*x^6 + 27662*x^7 + 238209*x^8 + 2176591*x^9 + 20928935*x^10 +...
1/A(x*A(x)) = 1 - x - 2*x^2 - 8*x^3 - 45*x^4 - 303*x^5 - 2293*x^6 - 18910*x^7 - 166921*x^8 - 1559040*x^9 - 15286286*x^10 +...
where A(x) = -1 + x*A(x) + A(x*A(x)) + 1/A(x*A(x)).

Prog

(PARI) {a(n)=local(A=[1, 1]); for(m=2, n+1, A[m]=Vec((-1+x+ Ser(A) +1/Ser(A))^m)[m]/m; A=concat(A, 0)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A241599 A356118 A307696 * A117399 A145345 A212027

Adjacent sequences: A237642 A237643 A237644 * A237646 A237647 A237648

Keyword

nonn

Author

Paul D. Hanna, May 02 2014

Status

approved