A212411 G.f. satisfies: A(x) = 1 + x*A(1 - 1/A(x))^2.

1, 1, 2, 7, 36, 235, 1792, 15261, 141382, 1401334, 14694166, 161714217, 1857003186, 22152227989, 273573165626, 3488210643709, 45820081884234, 618950367384072, 8585324020132250, 122127635117014779, 1779763238159032068, 26545963246376545934

Offset

0,3

Comments

Compare g.f. to the identity: G(x) = 1 + x*G(1-1/G(x)) when G(x) = 1/(1-x).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Formula

Given g.f. A(x), let G(x) be the g.f. of A213591, then:

(1) G(x) = x*A(G(x)^2/x),

(2) A( x/(1 - G(x)^2/x) ) = 1/(1 - G(x)^2/x),

(3) x = G(x - G(x)^2).

G.f.: A(x) = x/Series_Reversion(F(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628.

Given g.f. A(x), let F(x) = A(1-1/A(x)), then F(1-1/A(x)) = A(1-1/F(x)) and A(x) = 1 + x*F(x)^2.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 90*x^4 + 570*x^5 + 4247*x^6 +...
1 - 1/A(x) = x + x^2 + 4*x^3 + 23*x^4 + 161*x^5 + 1286*x^6 + 11321*x^7 +...
A(1-1/A(x)) = 1 + x + 3*x^2 + 15*x^3 + 98*x^4 + 753*x^5 + 6471*x^6 +...
Let F(x) = A(1-1/A(x)), then F(1-1/A(x)) = A(1-1/F(x)):
F(1-1/A(x)) = 1 + x + 4*x^2 + 25*x^3 + 193*x^4 + 1693*x^5 + 16240*x^6 +...
...
Let G(x) be the g.f. of A213591, then
G(x) satisfies: x = G(x - G(x)^2) and G(x) = A(G(x)^2/x), where:
G(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
G(x)^2/x = x + 2*x^2 + 9*x^3 + 56*x^4 + 420*x^5 + 3572*x^6 +...
1/(1-G(x)^2/x) = 1 + x + 3*x^2 + 14*x^3 + 85*x^4 + 616*x^5 + 5072*x^6 +...
such that A(x/(1 - G(x)^2/x)) = 1/(1 - G(x)^2/x).

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*subst(A^2, x, 1-1/(A+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A213591, A213628.

Sequence in context: A018997 A018954 A019030 * A119736 A353166 A308750

Adjacent sequences: A212408 A212409 A212410 * A212412 A212413 A212414

Keyword

nonn

Author

Paul D. Hanna, May 15 2012

Status

approved