A380678 G.f. A(x) satisfies A( x - A(x)^2/(1 - A(x)^2) ) = x.

1, 1, 4, 25, 190, 1645, 15652, 160186, 1739032, 19838179, 236192158, 2920269202, 37352521348, 492799406899, 6690428699026, 93293086422514, 1334088426585850, 19538994465481000, 292775222237716612, 4484180296611470218, 70146488080451823382, 1119964903188050808163, 18239593214541431577550

Offset: 1

Author

Paul D. Hanna, Feb 14 2025

Keywords

• nonn

Comments

Conjecture: for n>= 1, a(n) is odd iff n = A027383(k) for k >= 0, where A027383(2*m) = 3*2^m - 2 and A027383(2*m+1) = 4*2^m - 2.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1645*x^6 + 15652*x^7 + 160186*x^8 + 1739032*x^9 + 19838179*x^10 + ...
where A(x - A(x)^2/(1 - A(x)^2)) = x.
Let B(x) = x - Series_Reversion(A(x)), where
B(x) = x^2 + 2*x^3 + 10*x^4 + 62*x^5 + 469*x^6 + 4028*x^7 + 37984*x^8 + 385202*x^9 + 4144798*x^10 + ... + A380558(n)*x^n + ...
then A(x) = x + B(A(x)) and B(x - B(x)) = x^2/(1 - x^2).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..520

Crossrefs

Cf. A380558, A213591, A027383.

Programs

PARI

/* Generates N terms of this sequence */
N = 40; A=x; for(m=1,N, A=truncate(A); A = serreverse(x - A^2/(1 - A^2 +O(x^m)) ); ); Vec(A)

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A( x - A(x)^2/(1 - A(x)^2) ) = x.

(2) A(x) = x + A(A(x))^2/(1 - A(A(x))^2).

(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/(1 - A(x)^2)^n / n!.

(4) A(x) = x + B(A(x)) and A(x - B(x)) = x where B(x) = A(x)^2/(1 - A(x)^2) is the g.f. of A380558.