A091561 Expansion of (1-2x-sqrt(1-4x+4x^2-4x^3))/(2x^2).

1, 2, 4, 9, 22, 56, 146, 388, 1048, 2869, 7942, 22192, 62510, 177308, 506008, 1451866, 4185788, 12119696, 35227748, 102753800, 300672368, 882373261, 2596389190, 7658677856, 22642421206, 67081765932, 199128719896, 592179010350

Offset

1,2

Links

Table of n, a(n) for n=1..28.

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Formula

G.f.: (1-2x-sqrt(1-4x+4x^2-4x^3))/(2x^2).

a(n) = 2*a(n-1)+a(1)*a(n-3)+a(2)*a(n-4)+...+a(n-3)*a(1) for n>1.

Series reversion of g.f. A(x) is -A(-x).

G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=(xy)^2+2(xy)-(y-x).

Conjecture: (n+2)*a(n) -2*(2*n+1)*a(n-1) +4*(n-1)*a(n-2) +2*(5-2*n)*a(n-3)=0. - R. J. Mathar, Aug 14 2012

Mathematica

CoefficientList[Series[(1-2x-Sqrt[1-4x+4x^2-4x^3])/(2x^2), {x, 0, 30}], x] (* Harvey P. Dale, Jan 31 2015 *)

Prog

(PARI) a(n)=polcoeff((1-2*x-sqrt(1-4*x+4*x^2-4*x^3+x^3*O(x^n)))/2, n+2)

Crossrefs

Cf. A025247, A025265.

Sequence in context: A352702 A055588 A088456 * A025265 A152225 A037245

Adjacent sequences: A091558 A091559 A091560 * A091562 A091563 A091564

Keyword

nonn

Author

Michael Somos, Jan 20 2004

Status

approved