|
%I #25 Sep 15 2021 05:57:41
%S 1,5,24,113,526,2430,11166,51105,233190,1061510,4822984,21879786,
%T 99135076,448707992,2029215114,9170247393,41416383366,186957126702,
%U 843575853984,3804927658878,17156636097156,77339426905812
%N Expansion of 1/(sqrt(1-4*x)*(1-3*x*c(x))), where c(x) is the g.f. of A000108.
%C Binomial transform of A126932. Hankel transform is (-1)^n.
%C 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
%H Michael De Vlieger, <a href="/A141223/b141223.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%F a(n) = Sum_{k=0..n} C(2n-k,n-k)*3^k.
%F From _Emanuele Munarini_, Apr 01 2011: (Start)
%F a(n) = [x^n] 1/((1+x)^(n+1)*(1-3x)).
%F a(n) = 3^(2n+1)/2^(n+2) + (1/4)*sum(binomial(2k,k)*(9/2)^(n-k),k=0..n).
%F D-finite with recurrence: 2*(n+2)*a(n+2) - (17*n+30)*a(n+1) + 18*(2*n+3)*a(n) = 0.
%F G.f.: (3-12*x+sqrt(1-4*x))/(4-34*x+72*x^2). (End)
%F 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
%F a(n) ~ 3^(2*n + 1) / 2^(n + 1). - _Vaclav Kotesovec_, Sep 15 2021
%t CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2),{x,0,100}],x] (* _Emanuele Munarini_, Apr 01 2011 *)
%o (Maxima) makelist(sum(binomial(n+k,k)*3^(n-k),k,0,n),n,0,12); /* _Emanuele Munarini_, Apr 01 2011 */
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jun 14 2008
|
