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A084772
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Coefficients of 1/sqrt(1 - 12*x + 16*x^2); also, a(n) is the central coefficient of (1 + 6*x + 5*x^2)^n.
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4
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1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556, 3038019876, 30354866856, 304523343996, 3065412858696, 30946859111256, 313206733667376, 3176825392214406, 32284147284682596, 328643023505612596
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OFFSET
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0,2
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COMMENTS
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Diagonal of rational functions 1/(1 - x - y - 4*x*y), 1/(1 - x - y*z - 4*x*y*z). - Gheorghe Coserea, Jul 06 2018
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LINKS
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FORMULA
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E.g.f.: exp(6*x)*BesselI(0, 2*sqrt(5)*x). - Paul Barry, Sep 20 2004
Asymptotic: a(n) ~ (1+sqrt(5))^(2*n+1)/(2*5^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 11 2012
D-finite with recurrence: n*a(n) = 6*(2*n-1)*a(n-1) - 16*(n-1)*a(n-2). - R. J. Mathar, Nov 09 2012
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EXAMPLE
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G.f.: 1/sqrt(1 - 2*b*x + (b^2-4*c)*x^2) yields central coefficients of (1 + b*x + c*x^2)^n.
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MATHEMATICA
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Table[n! SeriesCoefficient[E^(6 x) BesselI[0, 2 Sqrt[5] x], {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, May 10 2013 *)
CoefficientList[Series[1/Sqrt[1-12x+16x^2], {x, 0, 30}], x] (* Harvey P. Dale, Apr 17 2015 *)
Table[4^n*LegendreP[n, 3/2], {n, 0, 40}] (* G. C. Greubel, May 31 2023 *)
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PROG
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(PARI) for(n=0, 30, t=polcoeff((1+6*x+5*x^2)^n, n, x); print1(t", "))
(GAP) List([0..20], n->Sum([0..n], k->Binomial(n, k)^2*5^k)); # Muniru A Asiru, Jul 29 2018
(Magma) [4^n*Evaluate(LegendrePolynomial(n), 3/2) : n in [0..40]]; // G. C. Greubel, May 31 2023
(SageMath) [4^n*gen_legendre_P(n, 0, 3/2) for n in range(41)] # G. C. Greubel, May 31 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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