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A082305
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G.f.: (1 - 6*x - sqrt(36*x^2 - 16*x + 1))/(2*x).
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8
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1, 7, 56, 497, 4760, 48174, 507696, 5516133, 61363736, 695540258, 8004487568, 93283238986, 1098653880688, 13056472392796, 156371970692448, 1885491757551213, 22870028390806296, 278862330338622618
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OFFSET
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0,2
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COMMENTS
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More generally coefficients of (1 - m*x - sqrt(m^2*x^2 - (2*m+4)*x + 1))/(2*x) are given by a(0)=1 and a(n) = (1/n)*Sum_{k=0..n} (m+1)^k * C(n,k) *C(n,k-1) for n > 0.
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LINKS
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FORMULA
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a(n) = (1/n)*Sum_{k=0..n} 7^k*C(n, k)*C(n, k-1), a(0)=1.
D-finite with recurrence: (n+1)*a(n) + 8*(1-2*n)*a(n-1) + 36*(n-2)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ sqrt(14+8*sqrt(7))*(8+2*sqrt(7))^n*(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - 6*x - x/(1 - 6*x - x/(1 - 6*x - x/(1 - 6*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 04 2018
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MATHEMATICA
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Table[SeriesCoefficient[(1-6*x-Sqrt[36*x^2-16*x+1])/(2*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
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PROG
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(PARI) a(n)=if(n<1, 1, sum(k=0, n, 7^k*binomial(n, k)*binomial(n, k-1))/n)
(PARI) x='x+O('x^99); Vec((1-6*x-(36*x^2-16*x+1)^(1/2))/(2*x)) \\ Altug Alkan, Apr 04 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-6*x-Sqrt(36*x^2-16*x+1))/(2*x))); // G. C. Greubel, Sep 16 2018
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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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