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A115865
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a(n) = Legendre_P(n,2)*6^n.
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1
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1, 12, 198, 3672, 71766, 1444392, 29623644, 615614256, 12918175974, 273112332552, 5808412280628, 124127223181776, 2663248527920124, 57334738304731536, 1237861064261885688, 26791929483836768352
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OFFSET
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0,2
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COMMENTS
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Central coefficients of (1+12*x+27*x^2)^n. In general, Jacobi_P(n,0,0,sqrt(m))(k*sqrt(m))^n = Legendre_P(n,sqrt(m))(k*sqrt(m))^n has g.f. 1/sqrt(1-2*k*m*x+k^2*x^2), e.g.f. exp(k*m*x)Bessel_I(0,k*sqrt(m(m-1))*x) and gives the central coefficients of (1+k*m*x+k^2*(m(m-1)/4)*x^2)^n.
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LINKS
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FORMULA
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G.f.: 1/sqrt(1-24*x+36*x^2).
E.g.f.: exp(12*x)*Bessel_I(0,3*sqrt(12)x).
a(n) = Jacobi_P(n,0,0,sqrt(4))*(3*sqrt(4))^n.
D-finite with recurrence: n*a(n) +12*(1-2*n)*a(n-1) +36*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ sqrt(18+12*sqrt(3))*(12+6*sqrt(3))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1-24*x+36*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
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PROG
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(PARI) x='x+O('x^50); Vec(1/sqrt(1-24*x+36*x^2)) \\ G. C. Greubel, Mar 18 2017
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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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STATUS
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approved
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