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%I #25 Nov 08 2022 01:47:16
%S 1,1,10,19,190,442,4420,11395,113950,312814,3128140,8960878,89608780,
%T 264735892,2647358920,8006545891,80065458910,246643289830,
%U 2466432898300,7711583225338,77115832253380,244082045341036,2440820453410360,7805301802531534
%N Generalized central coefficients for k=3.
%C Hankel transform of a(n) is 9^binomial(n+1,2). Case k=3 of T(n,k) = (1/Pi)*2*k^2*(2*k)^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(1+k^2-2*k*x) dx. T(n,k) has Hankel transform (k^2)^binomial(n+1,2). k=1 corresponds to C(n, floor(n/2)).
%C Expansion of c(9*x^2)/(1-x*c(9*x^2)), where c(x) is the g.f. of A000108. Reversion of x*(1+x)/(1+2*x+10*x^2). - _Philippe Deléham_, Nov 09 2007
%H Vincenzo Librandi, <a href="/A121725/b121725.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = (1/Pi)*18*6^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(10-6*x) dx.
%F a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*3^2k. - _Philippe Deléham_, Aug 18 2006
%F a(n) = Sum_{k=0..n} A120730(n,k)*9^(n-k). - _Philippe Deléham_, Nov 09 2007
%F Conjecture: (n+1)*a(n) = 10*(n+1)*a(n-1) + 36*(n-2)*a(n-2) - 360*(n-2)*a(n-3). - _R. J. Mathar_, Nov 26 2012
%F From _Vaclav Kotesovec_, Feb 13 2014: (Start)
%F a(n) ~ (4+(-1)^n) * 2^(n-7/2) * 3^(n+2) / (n^(3/2) * sqrt(Pi)).
%F G.f.: (1 - sqrt(1 - 36*x^2))/(18*x^2 - x*(1 - sqrt(1 - 36*x^2))). (End)
%t CoefficientList[Series[(1-Sqrt[1-4*9*x^2])/(2*9*x^2-x*(1-Sqrt[1-4*9*x^2])), {x,0,40}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-36*x^2))/(18*x^2-x*(1-Sqrt(1-36*x^2))) )); // _G. C. Greubel_, Nov 07 2022
%o (SageMath)
%o def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
%o def A121725(n): return sum(9^(n-k)*A120730(n,k) for k in range(n+1))
%o [A121725(n) for n in range(41)] # _G. C. Greubel_, Nov 07 2022
%Y Cf. A000108, A009766, A120730.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Aug 17 2006
%E More terms from _Vincenzo Librandi_, Feb 15 2014
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