A120010 - OEIS

%I #23 Dec 08 2021 11:26:47

%S 1,1,1,2,6,18,53,158,481,1491,4688,14913,47913,155261,506881,1665643,

%T 5504988,18287338,61027991,204499397,687808931,2321177071,7857504876,

%U 26673769002,90783820081,309720079813,1058984020333,3628267267358

%N G.f.: A(x) = (1-sqrt(1-4*x))/2 o x/(1-x) o (x-x^2), a composition of functions involving the Catalan function and its inverse.

%C The n-th iteration of g.f. A(x) is: (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) = (1 - sqrt(1 - 4*(x-x^2)/(1-n*x+n*x^2) ))/2. See A120009 for the transpose of the composition of the same functions.

%C Row sums of A155839. [_Paul Barry_, Jan 28 2009]

%H Vincenzo Librandi, <a href="/A120010/b120010.txt">Table of n, a(n) for n = 1..200</a>

%H S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt"> Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017)

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

%F a(n)=sum{k=0..n, sum{j=0..n, (-1)^(n-j)*C(j+1,n-j)*C(j,k)*if(k<=j, A000108(j-k),0)}}. [offset 0]. [_Paul Barry_, Jan 28 2009]

%F Conjecture: n*a(n) +2*(4-3*n)*a(n-1) +(11*n-26)*a(n-2) +10*(3-n)*a(n-3) +5*(n-4)*a(n-4)= 0. - _R. J. Mathar_, Nov 14 2011

%F a(n) ~ sqrt(5*sqrt(5)-5) * (5+sqrt(5))^n / (sqrt(Pi) * n^(3/2) * 2^(n+7/2)). - _Vaclav Kotesovec_, Feb 13 2014

%F Equivalently, a(n) ~ 5^((n+1)/2) * phi^(n - 1/2) / (8 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 08 2021

%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +...

%t Rest[CoefficientList[Series[(1-Sqrt[1-4*(x-x^2)/(1-x+x^2)])/2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Feb 13 2014 *)

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

%o for(n=1,35,print1(a(n),", "))

%Y Cf. A120009 (composition transpose), A000108 (Catalan).

%K nonn

%O 1,4

%A _Paul D. Hanna_, Jun 03 2006