A141223 Expansion of 1/(sqrt(1-4*x)*(1-3*x*c(x))), where c(x) is the g.f. of A000108.

1, 5, 24, 113, 526, 2430, 11166, 51105, 233190, 1061510, 4822984, 21879786, 99135076, 448707992, 2029215114, 9170247393, 41416383366, 186957126702, 843575853984, 3804927658878, 17156636097156, 77339426905812

Offset

0,2

Comments

Binomial transform of A126932. Hankel transform is (-1)^n.

Row sums of the Riordan matrix (1/(1-4*x),(1-sqrt(1-4*x))/(2*sqrt(1-4*x)) (A188481). - Emanuele Munarini, Apr 01 2001

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1000

Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Formula

a(n) = Sum_{k=0..n} C(2n-k,n-k)*3^k.

From Emanuele Munarini, Apr 01 2011: (Start)

a(n) = [x^n] 1/((1+x)^(n+1)*(1-3x)).

a(n) = 3^(2n+1)/2^(n+2) + (1/4)*sum(binomial(2k,k)*(9/2)^(n-k),k=0..n).

D-finite with recurrence: 2*(n+2)*a(n+2) - (17*n+30)*a(n+1) + 18*(2*n+3)*a(n) = 0.

G.f.: (3-12*x+sqrt(1-4*x))/(4-34*x+72*x^2). (End)

G.f.: (1/(1-4*x)^(1/2)+3)/(4-18*x)=( 2 + x/(Q(0)-2*x))/(2-9*x) where Q(k) = 2*(2*k+1)*x + (k+1) - 2*(k+1)*(2*k+3)*x/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Mar 18 2013

a(n) ~ 3^(2*n + 1) / 2^(n + 1). - Vaclav Kotesovec, Sep 15 2021

Mathematica

CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2), {x, 0, 100}], x] (* Emanuele Munarini, Apr 01 2011 *)

Prog

(Maxima) makelist(sum(binomial(n+k, k)*3^(n-k), k, 0, n), n, 0, 12); /* Emanuele Munarini, Apr 01 2011 */

Crossrefs

Sequence in context: A218987 A272257 A347029 * A289783 A140766 A026388

Adjacent sequences: A141220 A141221 A141222 * A141224 A141225 A141226

Keyword

easy,nonn

Author

Paul Barry, Jun 14 2008

Status

approved