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