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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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16
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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". - Gary W. Adamson, Sep 14 2008
Starting (1, 1, 1, ...) = inverse binomial transform of A014137: (1, 2, 4, 9, 23, 65, ...). - Gary W. Adamson, Apr 02 2009
With a(0) = 1 this is the Riordan transform with the Riordan matrix R(n, m) = (-1)^(n-m)*A097805(n, m) (the inverse of A097805) of the Catalan sequence A000108. See a Feb 17 2017 comment on A097805 for Riordan transforms, and the g.f. given below in terms of the Catalan g.f. - Wolfdieter Lang, Feb 17 2017
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LINKS
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Joerg Arndt, Table of n, a(n) for n = 0..200
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Gi-Sang Cheon, Marshall M. Cohen and Nikolaos Pantelidis, Decompositions and eigenvectors of Riordan matrices, Linear Algebra and its Applications, Vol. 642 (2022), 118-138.
T. Feil, K. Hutson and 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 - 2*x - 3*x^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
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
a(n) ~ 3^(n-1/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 20 2014
a(n) = Sum_{k=1..n} binomial(2*k-2, k-1)*(-1)^(n-k)*binomial(n-2, n-k)/k. - Vladimir Kruchinin, May 27 2014
G.f. if a(0) = 1: C(x/(1+x)) with C the g.f. of A000108 (Catalan). See an above implicit formula. - Wolfdieter Lang, Feb 17 2017
D-finite with recurrence: (3*n-3)*a(n)+(1+2*n)*a(n+1)+(-n-2)*a(n+2)=0 for n >= 1. - Robert Israel, May 01 2018
a(n) = A007971(n)/2, n>=2. - R. J. Mathar, Jan 20 2020
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EXAMPLE
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G.f. = x + 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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MAPLE
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with(PolynomialTools): CoefficientList(convert(taylor((1 + x - sqrt(1 - 2*x - 3*x^2))/2, x = 0, 33), polynom), x); # Taras Goy, Aug 07 2017
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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 *)
a = DifferenceRoot[Function[{y, n}, {(3n-3)*y[n] + (2n+1)*y[n+1] + (-n-2)*y[n+2] == 0, y[0] == 0, y[1] == 1, y[2] == 1}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 28 2021 *)
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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)}
(PARI) x='x+O('x^99); concat(0, Vec((1+x-(1-2*x-3*x^2)^(1/2))/2)) \\ Altug Alkan, May 01 2018
(Maxima) a(n):=sum((binomial(2*k-2, k-1)*(-1)^(n-k)*binomial(n-2, n-k))/k, k, 1, n); /* Vladimir Kruchinin, May 27 2014 */
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CROSSREFS
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a(n+2) = A001006(n).
Cf. A000108, A014137, A144218.
Sequence in context: A292440 A168049 A001006 * A247100 A230556 A027057
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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