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A240880
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Expansion of g.f.: (-1 + sqrt(1+12*x+48*x^2)) / (6*x).
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3
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1, 1, -6, 33, -162, 666, -1836, -2079, 79542, -741474, 4907628, -24837030, 82449900, 53319060, -3741922008, 38613958497, -274566158298, 1475669401398, -5211777090564, -2356585871778, 240686500011588, -2593621485808596, 19047621883804056, -105353643788834598
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OFFSET
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0,3
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COMMENTS
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This sequence is the member (q=-3) of a class of generalized Catalan numbers (see A000108), with g.f. (1-sqrt(1-q*4*x*(1-(q-1)*x)))/(2*q*x), q<>0.
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LINKS
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FORMULA
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G.f.: (-1 + sqrt(1+12*x+48*x^2)) / (6*x).
D-finite with recurrence: (n+3)*a(n+2)+6*(2*n+3)*a(n+1)+48*n*a(n)=0, a(0)=1, a(1)=1.
Lim sup n->infinity |a(n)|^(1/n) = 4*sqrt(3) = 6.9282... - Vaclav Kotesovec, May 02 2014
a(n) ~ 3^(n/2-1)*4^n / (n^(3/2)*sqrt(Pi)) * (sqrt(3)*cos(5*Pi*n/6) + 3*sin(5*Pi*n/6) - (15*sqrt(3)*cos(5*Pi*n/6) + 9*sin(5*Pi*n/6))/(8*n)). - Vaclav Kotesovec, May 02 2014
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MATHEMATICA
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CoefficientList[Series[(Sqrt[1+12x+48x^2]-1)/(6x), {x, 0, 30}], x] (* Harvey P. Dale, May 24 2022 *)
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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