login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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.
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-François 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).
Sequence in context: A292440 A168049 A001006 * A247100 A230556 A027057
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 13 2003
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)