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A157004
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Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.
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5
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1, 2, 6, 18, 58, 192, 650, 2232, 7746, 27096, 95376, 337404, 1198546, 4272308, 15273888, 54744268, 196646922, 707747988, 2551624304, 9213416524, 33313656888, 120604436624, 437112790668, 1585877246424, 5759085911154
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OFFSET
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0,2
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COMMENTS
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Hankel transform is A157005. Image of A000984 under Riordan array (1,x(1-x^2)).
Diagonal of rational function 1/(1 - x - y + x^3*y^2). - Seiichi Manyama, Mar 23 2023
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LINKS
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FORMULA
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G.f.: 1/sqrt(1 - 4*x*(1 - x^2)).
a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2)) *A000984(k)/2.
G.f.: Sum_{n>=0} (2*n)!/n!^2 * x^(2*n) * (1-x)^n / (1-2*x)^(2*n+1). - Paul D. Hanna, Sep 21 2013
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 2*(2*n-3)*a(n-3). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ (1/r)^n / (sqrt(Pi*n) * sqrt(3-8*r)), where r = 0.2695944364054... is the root of the equation 4*r*(1-r^2)=1. - Vaclav Kotesovec, Feb 13 2014
0 = a(n)*(16*a(n+1) - 32*a(n+3) + 10*a(n+4)) + a(n+1)*(-2*a(n+3)) + a(n+2)*(16*a(n+3) - 6*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Sep 03 2016
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EXAMPLE
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G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 192*x^5 + 650*x^6 + 2232*x^7 + ...
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1-4*x*(1-x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
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PROG
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(PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m)*(1-x)^m / (1-2*x+x*O(x^n))^(2*m+1)), n)} \\ Paul D. Hanna, Sep 21 2013
(PARI) my(x='x+O('x^30)); Vec(1/sqrt(1-4*x+4*x^3)) \\ G. C. Greubel, Feb 26 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-4*x+4*x^3) )); // G. C. Greubel, Feb 26 2019
(Sage) (1/sqrt(1-4*x+4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019
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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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