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A086246 Expansion of (1 + x - sqrt(1 - 2*x - 3*x^2)) / 2 in powers of x. 9
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". [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]

LINKS

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.

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

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 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 2 x - 3 x^2])/2, {x, 0, n}] (* Michael Somos, Jan 25 2014 *)

PROG

(PARI) {a(n) = polcoeff( (1 + x - sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2, n)}

(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. A144218 [From Gary W. Adamson, Sep 14 2008]

A014137 [From Gary W. Adamson, Apr 02 2009]

Sequence in context: A166587 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 December 10 11:53 EST 2016. Contains 279001 sequences.