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%I #28 Dec 26 2023 16:30:51
%S 1,-1,2,-3,7,-12,30,-55,143,-273,728,-1428,3876,-7752,21318,-43263,
%T 120175,-246675,690690,-1430715,4032015,-8414640,23841480,-50067108,
%U 142498692,-300830572,859515920,-1822766520,5225264024,-11124755664,31983672534,-68328754959
%N Expansion of reversion of (x - 2*x^2) / (1 - x)^3.
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%F Given g.f. A(x), then 1 = (1/A(x) + 1/A(-x)) / 2.
%F a(n) = -(-1)^n * binomial(n + m, n - m) / (2*m + 1) where m = floor(n/2) if n>0.
%F From _Michael Somos_, Apr 13 2012 (Start)
%F a(n) = -(-1)^n * A047749(n) unless n=0. a(2*n) = - A001764(n) unless n=0. a(2*n + 1) = A006013(n).
%F Reversion of A080956 with offset 1.
%F Hankel transform is A005161 omitting first 1.
%F n * a(n) = -(-1)^n * A099576(n-1). (End)
%F D-finite with recurrence +8*n*(n+1)*a(n) -36*n*(n-2)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - _R. J. Mathar_, Sep 24 2021
%F a(n) = (-1)^(n-1)*binomial(2*n, n-1)*hypergeom([-n+1, n], [-2*n], -1) / n. - _Detlef Meya_, Dec 26 2023
%e G.f. = x - x^2 + 2*x^3 - 3*x^4 + 7*x^5 - 12*x^6 + 30*x^7 - 55*x^8 + 143*x^9 + ...
%t a[ n_] := With[ {m = Quotient[n, 2]}, If[n < 1, 0, -(-1)^n Binomial[n + m, n - m] / (2 m + 1)]]; (* _Michael Somos_, Oct 16 2015 *)
%t a[ n_] := If[n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[(x - 2 x^2) / (1 - x)^3, {x, 0, n}]], {x, 0, n}]]; (* _Michael Somos_, Oct 16 2015 *)
%t a[n_] := (-1)^(n-1)*Binomial[2*n, n-1]*Hypergeometric2F1[-n+1, n, -2*n, -1] / n; Flatten[Table[a[n], {n, 1, 32}]] (* _Detlef Meya_, Dec 26 2023 *)
%o (PARI) {a(n) = my( m = n\2); if( n<1, 0, -(-1)^n * binomial( n + m, n - m) / (2 * m + 1))};
%o (PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( (x - 2 * x^2) / (1 - x)^3 + x * O(x^n) ), n))};
%o (PARI) {a(n) = if( n<1, 0, polcoeff( 1 / ( 1 + 1 / serreverse( x - x^3 + x * O(x^n) )), n))};
%Y Cf. A001764, A005161, A006013, A047749, A080956, A099576.
%K sign
%O 1,3
%A _Michael Somos_, Nov 01 2007