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A086246
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Expansion of (1+x-sqrt(1-2*x-3*x^2))/2 in powers of x.
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7
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0, 1, 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, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391, 13933569346707
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OFFSET
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0,5
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COMMENTS
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A variant of the Motzkin numbers: see A001006 for the main entry.
Equals row sums of triangle A144218 starting with "1". [From Gary W. Adamson, Sep 14 2008]
Starting (1, 1, 1,...) = inverse binomial transform of A014137: (1, 2, 4, 9, 23, 65,...). [From Gary W. Adamson, Apr 02 2009]
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LINKS
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Joerg Arndt, Table of n, a(n) for n = 0..200
T. Feil, K. Hutson, R. M. Kretchmar, Tree Traversals and Permutations, Congr. Numer. (2005), omitting the leading 0 and with a typo in the last number (303 should be 323), last sentence of chapter 6.
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FORMULA
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Series reversion of g.f. A(x) is -A(-x).
a(n)+a(n-1)=a(0)*a(n)+a(1)*a(n-1)+...+a(n)*a(0), n>2.
G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=x-y-x*y+x^2+y^2.
G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=(y^2-y^3)-(x^2+x^3).
G.f.: (1+x-sqrt(1-2x-3x^2))/2.
G.f. A(x) satisfies A(x) = x+C(x*A(x)) where C(x) is g.f. for Catalan numbers A000108 (offset 1).
G.f.: (1+x-sqrt(1-2*x-3*x^2))/2 = (x+x/G(0))/2 where G(k) = 1-2*x/(1+x/(1+x/(1-2*x/(1-x/(2-x/G(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011
G.f.: x + x^2*Q(0), where Q(k)= 1 + x/(1 - x - x/(x + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 25 2013
G.f.: x*Q(0), where Q(k)= 1 + (4*k+1)*x/((1+x)*(k+1) - x*(1+x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1+x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
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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)
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CROSSREFS
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a(n+2)=A001006(n).
Cf. A144218 [From Gary W. Adamson, Sep 14 2008]
A014137 [From Gary W. Adamson, Apr 02 2009]
Sequence in context: A166587 A168049 A001006 * A027057 A148071 A000636
Adjacent sequences: A086243 A086244 A086245 * A086247 A086248 A086249
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Jul 13 2003
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STATUS
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approved
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