A380558 G.f. A(x) satisfies A(x - A(x)) = x^2/(1 - x^2).

1, 2, 10, 62, 469, 4028, 37984, 385202, 4144798, 46882400, 553733875, 6795347708, 86314711993, 1131422763410, 15268625617174, 211726229534738, 3012057754693912, 43903115899714844, 654923002676505376, 9989373316478767304, 155663132037403882606, 2476418549848925209424, 40195761790035415573939

Offset

2,2

Comments

Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.

Links

Paul D. Hanna, Table of n, a(n) for n = 2..520

Formula

G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.

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

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

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

Example

G.f.: A(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 + 46882400*x^11 + 553733875*x^12 + ...
where A(x - A(x)) = x^2/(1 - x^2).
Let B(x) = Series_Reversion(x - A(x)), where
B(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 + ... + A380678(n)*x^n + ...
then B(x) = x + A(B(x)).

Prog

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

Crossrefs

Cf. A380678, A276370, A004760.

Sequence in context: A307364 A141140 A232472 * A361829 A361494 A371546

Adjacent sequences: A380555 A380556 A380557 * A380559 A380560 A380561

Keyword

nonn,new

Author

Paul D. Hanna, Feb 13 2025

Status

approved