A120010 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.

1, 1, 1, 2, 6, 18, 53, 158, 481, 1491, 4688, 14913, 47913, 155261, 506881, 1665643, 5504988, 18287338, 61027991, 204499397, 687808931, 2321177071, 7857504876, 26673769002, 90783820081, 309720079813, 1058984020333, 3628267267358

Offset: 1

Author

Paul D. Hanna, Jun 03 2006

Keywords

• nonn

Comments

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.

Row sums of A155839. [Paul Barry, Jan 28 2009]

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

Crossrefs

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

Programs

Mathematica

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

PARI

{a(n)=polcoeff((1 - sqrt(1 - 4*(x-x^2)/(1-x+x^2+x*O(x^n)) ))/2,n)}
for(n=1,35,print1(a(n),", "))

Formula

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

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]

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

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

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