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A240688
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Expansion of -(x*sqrt(-4*x^2-4*x+1)-2*x^2-3*x) / ((x+1)*sqrt(-4*x^2-4*x+1)+ 4*x^3+8*x^2+3*x-1).
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4
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1, 1, 5, 19, 81, 351, 1553, 6959, 31489, 143551, 658305, 3033471, 14034177, 65147135, 303285505, 1415422719, 6620053505, 31021657087, 145613977601, 684537354239, 3222408929281, 15187861143551, 71663163121665
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} Sum_{i=0..(n-k)} binomial(k,n-k-i)*binomial(k+i-1,i)*binomial(n,k).
A(x) = x*D'(x)/D(x) where D(x)=(1-sqrt(1-4*x-4*x^2))/(2*(1+x)) is g.f. of A052709.
a(n) ~ 2^(n-1/4) * (1+sqrt(2))^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Apr 12 2014
a(n) = Sum_{i=0..n/2} binomial(n,i)*binomial(2*n-2*i-1,n-2*i). - Vladimir Kruchinin, Mar 10 2015
Conjecture: n*(n-1)*a(n) -(3*n-2)*(n-1)*a(n-1) +2*(-4*n^2+7*n-1)*a(n-2) -4*n*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 14 2016
The o.g.f. A(x) satisfies the differential equation (8*x^4 + 20*x^3 + 14*x^2 + x - 1)*A(x)' + (8*x^3 + 12*x^2 + 6*x + 3)*A(x) - 2 = 0 with A(0) = 1.
n*a(n) = (n+2)*a(n-1) + (14*n-22)*a(n-2) + (20*n-48)*a(n-3) + (8*n-24)*a(n-4).
Mathar's conjectural third-order recurrence above can be verified using Maple's gfun:-rectodiffeq command.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)
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MATHEMATICA
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CoefficientList[Series[-(x Sqrt[-4 x^2 - 4 x + 1] - 2 x^2 - 3 x) / ((x + 1) Sqrt[-4 x^2 - 4 x + 1] + 4 x^3 + 8 x^2 + 3 x - 1), {x, 0, 25}], x] (* Vaclav Kotesovec, Apr 12 2014 *)
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PROG
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(Maxima)
a(n):=sum((sum(binomial(k, n-k-i)*binomial(k+i-1, i), i, 0, n-k))*binomial(n, k), k, 0, n);
(PARI) x='x+O('x^50); Vec(-(x*sqrt(-4*x^2-4*x+1)-2*x^2-3*x) / ((x+1)*sqrt(-4*x^2-4*x+1)+ 4*x^3+8*x^2+3*x-1)) \\ G. C. Greubel, Apr 05 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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