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A084773
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Coefficients of 1/sqrt(1-12*x+4*x^2); also, a(n) is the central coefficient of (1+6*x+8*x^2)^n.
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12
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1, 6, 52, 504, 5136, 53856, 575296, 6225792, 68026624, 748832256, 8291791872, 92255680512, 1030537089024, 11550176206848, 129824329777152, 1462841567576064, 16518691986407424, 186887008999047168, 2117944490818011136, 24038305911245635584, 273199990096465494016
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OFFSET
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0,2
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COMMENTS
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Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H and D can choose from two colors each (or where one step is monochrome and the other two are bicolored). - Paul Barry, May 30 2005
2^n*P_n(3), where P_n is the n-th Legendre polynomial. 2^n*LegendreP(n,k) yields the central coefficients of (1 + 2*k*x + (k^2-1)*x^2)^n, with g.f. 1/sqrt(1 -4*k*x +4*x^2) and e.g.f. exp(2*k*x)*BesselI(0, 2*sqrt(k^2-1)*x). - Paul Barry, May 30 2005
Diagonal of rational functions 1/(1 - x - 2*y - 2*x*y), 1/(1 - x - 2*y*z - 2*x*y*z), 1/(1 - 2*x - y*z - 2*x*y*z). - Gheorghe Coserea, Jul 07 2018
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LINKS
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FORMULA
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E.g.f.: exp(6*x)*Bessel_I(0, 2*sqrt(8)*x).
a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2(n-k), n)*(-1)^k*3^(n-2*k). (End)
D-finite with recurrence: n*a(n) + 6*(-2*n+1)*a(n-1) + 4*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - x*(6-2*x)*(2*k+1)/(x*(6-2*x)*(2*k+1) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013
a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 1) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 11 2018
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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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MAPLE
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a := n -> 2^n*hypergeom([-n, -n], [1], 2):
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[(1-12x+4x^2)], {x, 0, 20}], x] (* Harvey P. Dale, Dec 13 2017 *)
Table[2^n*LegendreP[n, 3], {n, 0, 40}] (* G. C. Greubel, May 21 2023 *)
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PROG
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(PARI) for(n=0, 30, t=polcoeff((1+6*x+8*x^2)^n, n, x); print1(t", "))
(Magma) [2^n*Evaluate(LegendrePolynomial(n), 3): n in [0..40]]; // G. C. Greubel, May 21 2023
(SageMath) [2^n*gen_legendre_P(n, 0, 3) for n in range(41)] # G. C. Greubel, May 21 2023
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CROSSREFS
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Sequences of the form 2^n*LegendreP(n, 2*m+1): A000079 (m=0), this sequence (m=1), A098270 (m=2).
See A152254 for another interpretation.
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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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