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A086246 Expansion of (1 + x - sqrt(1 - 2*x - 3*x^2)) / 2 in powers of x. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

LINKS

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.

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.

FORMULA

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

EXAMPLE

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 + ...

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Oct 28 2021 *)

PROG

(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 */

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 13 2003

STATUS

approved

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Last modified January 24 13:09 EST 2022. Contains 350538 sequences. (Running on oeis4.)