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A111961 Expansion of 1/(sqrt(1-2x-3x^2)-x). 3
1, 2, 6, 18, 56, 176, 558, 1778, 5686, 18230, 58558, 188366, 606588, 1955044, 6305418, 20347342, 65689088, 212146400, 685342218, 2214556478, 7157409064, 23136645472, 74801223162, 241863933094, 782131232390, 2529458676326 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums of A111960.

A transform of the Fibonacci numbers. - Paul Barry, Sep 23 2005

Apparently the Motzkin transform of (0 followed by A128588). [From R. J. Mathar, Dec 11 2008]

Inverse binomial transform of A026671. [From Philippe Deléham, Feb 11 2009]

Hankel transform is 2^n. [From Paul Barry, Mar 02 2010]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.

FORMULA

a(n)=sum{k=0..n, sum{j=0..n, C(n, j)*C((j-1)/2, (j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2}}

a(n)=sum{k=0..n, F(k+1)*sum{i=0..floor((n-k)/2), C(n, i)C(n-i, i+k)/(i+k+1)}}. - Paul Barry, Sep 23 2005

G.f.: M(x)^2/(2*M(x)-M(x)^2), where M(x) is the g.f. of the Motzkin numbers A001006; - Paul Barry, Feb 03 2006

G.f.: 1/(1-2x/(1-x/(1-x^2/(1-x/(1-x/91-x^2/(1-x/(1-x/(1-x^2/(1-... (continued fraction). [From Paul Barry, Mar 02 2010]

Conjecture: n*a(n) +(-4*n+3)*a(n-1) +3*(-n+1)*a(n-2) +2*(7*n-15)*a(n-3) +12*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 15 2012

a(n) ~ (1+sqrt(5))^n / sqrt(5). - Vaclav Kotesovec, Feb 08 2014

MATHEMATICA

CoefficientList[Series[1/(Sqrt[1-2*x-3*x^2]-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

CROSSREFS

Sequence in context: A291228 A091142 A275857 * A190861 A071721 A125306

Adjacent sequences:  A111958 A111959 A111960 * A111962 A111963 A111964

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Aug 23 2005

STATUS

approved

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Last modified November 14 06:14 EST 2018. Contains 317162 sequences. (Running on oeis4.)