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A141223 Expansion of 1/(sqrt(1-4*x)*(1-3*x*c(x))), where c(x) is the g.f. of A000108. 3
1, 5, 24, 113, 526, 2430, 11166, 51105, 233190, 1061510, 4822984, 21879786, 99135076, 448707992, 2029215114, 9170247393, 41416383366, 186957126702, 843575853984, 3804927658878, 17156636097156, 77339426905812 (list; graph; refs; listen; history; text; internal format)
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

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: A079028 A218987 A272257 * A289783 A140766 A026388

Adjacent sequences:  A141220 A141221 A141222 * A141224 A141225 A141226

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jun 14 2008

STATUS

approved

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Last modified August 5 07:54 EDT 2021. Contains 346464 sequences. (Running on oeis4.)