A134565 Expansion of reversion of (x - 2*x^2) / (1 - x)^3.

1, -1, 2, -3, 7, -12, 30, -55, 143, -273, 728, -1428, 3876, -7752, 21318, -43263, 120175, -246675, 690690, -1430715, 4032015, -8414640, 23841480, -50067108, 142498692, -300830572, 859515920, -1822766520, 5225264024, -11124755664, 31983672534, -68328754959

Offset

1,3

Links

Table of n, a(n) for n=1..32.

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Formula

Given g.f. A(x), then 1 = (1/A(x) + 1/A(-x)) / 2.

a(n) = -(-1)^n * binomial(n + m, n - m) / (2*m + 1) where m = floor(n/2) if n>0.

From Michael Somos, Apr 13 2012 (Start)

a(n) = -(-1)^n * A047749(n) unless n=0. a(2*n) = - A001764(n) unless n=0. a(2*n + 1) = A006013(n).

Reversion of A080956 with offset 1.

Hankel transform is A005161 omitting first 1.

n * a(n) = -(-1)^n * A099576(n-1). (End)

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

a(n) = (-1)^(n-1)*binomial(2*n, n-1)*hypergeom([-n+1, n], [-2*n], -1) / n. - Detlef Meya, Dec 26 2023

Example

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 + ...

Mathematica

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 *)
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 *)
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 *)

Prog

(PARI) {a(n) = my( m = n\2); if( n<1, 0, -(-1)^n * binomial( n + m, n - m) / (2 * m + 1))};
(PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( (x - 2 * x^2) / (1 - x)^3 + x * O(x^n) ), n))};
(PARI) {a(n) = if( n<1, 0, polcoeff( 1 / ( 1 + 1 / serreverse( x - x^3 + x * O(x^n) )), n))};

Crossrefs

Cf. A001764, A005161, A006013, A047749, A080956, A099576.

Sequence in context: A111759 A305751 A047749 * A300749 A100982 A186009

Adjacent sequences: A134562 A134563 A134564 * A134566 A134567 A134568

Keyword

sign

Author

Michael Somos, Nov 01 2007

Status

approved