A111961 Expansion of 1/(sqrt(1-2x-3x^2)-x).

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

Offset: 0

Author

Paul Barry, Aug 23 2005

Keywords

• easy
• nonn

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). - R. J. Mathar, Dec 11 2008

Inverse binomial transform of A026671. - Philippe Deléham, Feb 11 2009

Hankel transform is 2^n. - 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.
• Paul Barry, Moment sequences, transformations, and Spidernet graphs, arXiv:2307.00098 [math.CO], 2023.
• Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 17, 29.
• Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.

Crossrefs

Cf. A001006, A026671, A111960, A128588.

Programs

Mathematica

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

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). - Paul Barry, Mar 02 2010

D-finite with recurrence: 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