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

1,4

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]

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)

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

Example

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

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 *)

Prog

(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), ", "))

Crossrefs

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

Sequence in context: A005507 A252822 A094864 * A132790 A358417 A372899

Adjacent sequences: A120007 A120008 A120009 * A120011 A120012 A120013

Keyword

nonn

Author

Paul D. Hanna, Jun 03 2006

Status

approved