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A121725
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Generalized central coefficients for k=3.
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5
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1, 1, 10, 19, 190, 442, 4420, 11395, 113950, 312814, 3128140, 8960878, 89608780, 264735892, 2647358920, 8006545891, 80065458910, 246643289830, 2466432898300, 7711583225338, 77115832253380, 244082045341036, 2440820453410360, 7805301802531534
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OFFSET
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0,3
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COMMENTS
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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)).
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
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LINKS
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FORMULA
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a(n) = (1/Pi)*18*6^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(10-6*x) dx.
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
a(n) ~ (4+(-1)^n) * 2^(n-7/2) * 3^(n+2) / (n^(3/2) * sqrt(Pi)).
G.f.: (1 - sqrt(1 - 36*x^2))/(18*x^2 - x*(1 - sqrt(1 - 36*x^2))). (End)
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MATHEMATICA
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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 *)
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PROG
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(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
(SageMath)
def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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