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, 348553445817084, 1570548863858778, 7075531788285276
Offset: 0
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.
Programs
-
Mathematica
CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2),{x,0,100}],x] (* Emanuele Munarini, Apr 01 2011 *) -
Maxima
makelist(sum(binomial(n+k,k)*3^(n-k),k,0,n),n,0,12); /* Emanuele Munarini, Apr 01 2011 */
Formula
a(n) = Sum_{k=0..n} C(2*n-k,n-k)*3^k.
From Emanuele Munarini, Apr 01 2011: (Start)
a(n) = [x^n] 1/((1-x)^(n+1) * (1-3*x)). [Corrected by Seiichi Manyama, Aug 03 2025]
a(n) = 3^(2*n+1)/2^(n+2) + (1/4)*Sum_{k=0..n} binomial(2*k,k)*(9/2)^(n-k).
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
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k). - Seiichi Manyama, Aug 03 2025
a(n) = 3^(2*n+1)*2^(-n-1) - binomial(2*n+1, n)*(hypergeom([1, -1-n], [1+n], -1/2) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k). - Seiichi Manyama, Aug 07 2025
Comments