A201645 - OEIS

A201645

G.f.: x/sqrt(1 + x^2 - 2*x*sqrt(1 + 4*x^2)).

1, 1, 1, 3, 7, 11, 21, 53, 113, 211, 451, 1049, 2223, 4517, 9881, 22203, 47531, 100531, 220933, 489737, 1059137, 2284401, 5025959, 11088703, 24161133, 52644061, 115913011, 255469863, 559494883, 1226060651, 2702052381, 5957474213, 13092891293, 28792397139, 63518607791, 140165690233

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OFFSET

1,4

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

G.f. satisfies: A(-A(-x)) = x.

G.f.: A(x) = x/sqrt((1-x)^2 - 4*x^3*C(-x^2)) where C(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).

Let A^{n}(x) denote the n-th iteration of A(x), then:

(1) A^{n}(x) = x/sqrt(1 + n^2*x^2 - 2*n*x*sqrt(1 + 4*x^2));

(2) A^{n}(x) = x/sqrt(1-4*x^2) o x/(1-n*x) o x/sqrt(1+4*x^2), a composition of functions involving a g.f. of the central binomial coefficients (A000984) and its inverse.

a(n) ~ sqrt(3)*5^(n/2-1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 29 2013

D-finite with recurrence -(6*n-17)*(n-1)*(n-2)*a(n) +2*(-6*n^3+65*n^2-171*n+91)*a(n-2) +(n-4)*(138*n^2-625*n+545)*a(n-4) +60*(6*n-5)*(n-6)*(n-4)*a(n-6)=0. - R. J. Mathar, Nov 22 2024

EXAMPLE

G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 7*x^5 + 11*x^6 + 21*x^7 + 53*x^8 +...

where x^2/A(x)^2 = 1 - 2*x + x^2 - 4*x^3 + 4*x^5 - 8*x^7 + 20*x^9 - 56*x^11 + 168*x^13 -+... + (-1)^n*4*A000108(n)*x^(n+3) +...

The initial iterations of A(x) begin:

A(A(x)) = x + 2*x^2 + 4*x^3 + 12*x^4 + 40*x^5 + 124*x^6 + 384*x^7 +...,

A(A(x)) = x/sqrt(1 + 4*x^2 - 4*x*sqrt(1 + 4*x^2));

A(A(A(x))) = x + 3*x^2 + 9*x^3 + 33*x^4 + 135*x^5 + 561*x^6 + 2349*x^7 +...,

A(A(A(x))) = x/sqrt(1 + 9*x^2 - 6*x*sqrt(1 + 4*x^2));

A(A(A(A(x)))) = x + 4*x^2 + 16*x^3 + 72*x^4 + 352*x^5 + 1784*x^6 +...,

A(A(A(A(x)))) = x/sqrt(1 + 16*x^2 - 8*x*sqrt(1 + 4*x^2)).

Related expansion:

x/sqrt(1-4*x^2) = x + 2*x^3 + 6*x^5 + 20*x^7 + 70*x^9 + 252*x^11 +...+ A000984(n)*x^n +...