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 A201645 G.f.: x/sqrt(1 + x^2 - 2*x*sqrt(1 + 4*x^2)). 1

%I #16 May 27 2017 20:41:39

%S 1,1,1,3,7,11,21,53,113,211,451,1049,2223,4517,9881,22203,47531,

%T 100531,220933,489737,1059137,2284401,5025959,11088703,24161133,

%U 52644061,115913011,255469863,559494883,1226060651,2702052381,5957474213,13092891293,28792397139,63518607791,140165690233

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

%H G. C. Greubel, <a href="/A201645/b201645.txt">Table of n, a(n) for n = 1..1000</a>

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

%F 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).

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

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

%F (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.

%F a(n) ~ sqrt(3)*5^(n/2-1/2)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jun 29 2013

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

%e 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) +...

%e The initial iterations of A(x) begin:

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

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

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

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

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

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

%e Related expansion:

%e 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 +...

%t Rest[CoefficientList[Series[x/Sqrt[1 + x^2 - 2*x*Sqrt[1 + 4*x^2]], {x, 0, 50}], x]] (* _G. C. Greubel_, May 27 2017 *)

%o (PARI) {a(n)=polcoeff(x/sqrt(1 + x^2 - 2*x*sqrt(1 + 4*x^2 +x*O(x^n))),n)}

%Y Cf. A000984.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Dec 03 2011

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Last modified September 13 10:14 EDT 2024. Contains 375904 sequences. (Running on oeis4.)