A389476 G.f. A(x) satisfies: A( x - A(x)^2/(1 - A(x))^2 ) = x.

1, 1, 6, 47, 442, 4691, 54330, 672992, 8800734, 120416351, 1712882948, 25210553528, 382540272262, 5967422564959, 95485793149616, 1564404241900576, 26204548617091122, 448220637836513308, 7820801877411056668, 139085087620932828566, 2519186189511503436532, 46442459335479148708255

Offset

1,3

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n and the related series F(x) (the g.f. of A185897) satisfy 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) = F(x)/(1 + F(x)) where F(x - F(x)^2) = x/(1-x).

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

(5) F(x) = sqrt( x - Series_Reversion(A(x)) ) where F(x) = A(x)/(1 - A(x)) is the g.f. of A185897.

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

(7) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) F(x)^(2*n)/x / n! ), where F(x) = A(x)/(1 - A(x)).

Example

G.f.: A(x) = x + x^2 + 6*x^3 + 47*x^4 + 442*x^5 + 4691*x^6 + 54330*x^7 + 672992*x^8 + 8800734*x^9 + 120416351*x^10 + ...
where x = A( x - A(x)^2/(1 - A(x))^2 ).
RELATED SERIES.
Let F(x) be the g.f. of A185897, then
F(x) = A(x)/(1 - A(x)) = x + 2*x^2 + 9*x^3 + 64*x^4 + 574*x^5 + 5919*x^6 + 67205*x^7 + 820258*x^8 + ... + A185897(n)*x^n + ...
and F(x - F(x)^2) = x/(1-x).
Also,
F(x)^2 = A(x)^2/(1 - A(x))^2 = x^2 + 4*x^3 + 22*x^4 + 164*x^5 + 1485*x^6 + 15286*x^7 + 172514*x^8 + 2089350*x^9 + 26783526*x^10 + ...
where A(x) = x + F(A(x))^2.

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse(x - A^2/(1-A +x*O(x^n))^2)); polcoeff(A, n)}
for(n=1, 22, print1(a(n), ", "))

Crossrefs

Cf. A185897, A380678, A213591, A027383.

Sequence in context: A383558 A365186 A104256 * A394848 A289211 A307567

Adjacent sequences: A389473 A389474 A389475 * A389477 A389478 A389479

Keyword

nonn

Author

Paul D. Hanna, Oct 05 2025

Status

approved