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A168051
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Expansion of (1+x+sqrt(1-2x-3x^2))/2.
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5
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1, 0, -1, -1, -2, -4, -9, -21, -51, -127, -323, -835, -2188, -5798, -15511, -41835, -113634, -310572, -853467, -2356779, -6536382, -18199284, -50852019, -142547559, -400763223, -1129760415, -3192727797, -9043402501, -25669818476
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OFFSET
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0,5
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COMMENTS
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A signed variant of the Motzkin numbers A001006. Hankel transform is A168052.
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LINKS
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FORMULA
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D-finite with recurrence: n*a(n) -(2n-3)*a(n-1) -3*(n-3)*a(n-2)=0 if n>2. - R. J. Mathar, Dec 20 2011 [Edited by Michael Somos, Jan 25 2014]
0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 25 2014
G.f.: 1 + x - (x + x^2) / (1 + x - (x + x^2) / (1 + x - ...)). - Michael Somos, Mar 27 2014
G.f.: (1 + x + A(x)) / 2, where A(x) is the g.f. of A167022. (End)
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EXAMPLE
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G.f. = 1 - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 21*x^7 - 51*x^8 - 127*x^9 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (1 + x + Sqrt[1 - 2 x - 3 x^2]) / 2, {x, 0, n}] (* Michael Somos, Jan 25 2014 *)
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PROG
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(PARI) {a(n) = polcoeff( (1 + x + sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2, n)}; /* Michael Somos, Jan 25 2014 */
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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