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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.

%I A141223 #45 Aug 07 2025 08:34:38
%S A141223 1,5,24,113,526,2430,11166,51105,233190,1061510,4822984,21879786,
%T A141223 99135076,448707992,2029215114,9170247393,41416383366,186957126702,
%U A141223 843575853984,3804927658878,17156636097156,77339426905812,348553445817084,1570548863858778,7075531788285276
%N A141223 Expansion of 1/(sqrt(1-4*x)*(1-3*x*c(x))), where c(x) is the g.f. of A000108.
%C A141223 Binomial transform of A126932. Hankel transform is (-1)^n.
%C A141223 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 A141223 Michael De Vlieger, <a href="/A141223/b141223.txt">Table of n, a(n) for n = 0..1000</a>
%H A141223 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 A141223 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 A141223 a(n) = Sum_{k=0..n} C(2*n-k,n-k)*3^k.
%F A141223 From _Emanuele Munarini_, Apr 01 2011: (Start)
%F A141223 a(n) = [x^n] 1/((1-x)^(n+1) * (1-3*x)). [Corrected by _Seiichi Manyama_, Aug 03 2025]
%F A141223 a(n) = 3^(2*n+1)/2^(n+2) + (1/4)*Sum_{k=0..n} binomial(2*k,k)*(9/2)^(n-k).
%F A141223 D-finite with recurrence: 2*(n+2)*a(n+2) - (17*n+30)*a(n+1) + 18*(2*n+3)*a(n) = 0.
%F A141223 G.f.: (3-12*x+sqrt(1-4*x))/(4-34*x+72*x^2). (End)
%F A141223 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 A141223 a(n) ~ 3^(2*n + 1) / 2^(n + 1). - _Vaclav Kotesovec_, Sep 15 2021
%F A141223 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k). - _Seiichi Manyama_, Aug 03 2025
%F A141223 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
%F A141223 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
%t A141223 CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2),{x,0,100}],x] (* _Emanuele Munarini_, Apr 01 2011 *)
%o A141223 (Maxima) makelist(sum(binomial(n+k,k)*3^(n-k),k,0,n),n,0,12); /* _Emanuele Munarini_, Apr 01 2011 */
%Y A141223 Cf. A000108.
%Y A141223 Cf. A000302, A000984, A026641.
%Y A141223 Cf. A006256, A385605, A385632.
%K A141223 easy,nonn
%O A141223 0,2
%A A141223 _Paul Barry_, Jun 14 2008