A291534 Expansion of the series reversion of x/((1 + x)*(1 - x^2)).

1, 1, 0, -3, -7, -4, 24, 85, 99, -215, -1196, -2100, 1420, 17512, 42160, 9477, -252073, -815965, -736456, 3365813, 15248793, 22861712, -37036000, -273657748, -575046252, 180950476, 4658415696, 13042693000, 6717278152, -73400374512, -275797704864, -321427878811, 1012425395135

Offset

1,4

Comments

Reversion of g.f. for the canonical enumeration of integers (A001057).

Links

Table of n, a(n) for n=1..33.

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Series Reversion

Index entries for reversions of series

Formula

G.f. A(x) satisfies: A(x)/((1 + A(x))*(1 - A(x)^2)) = x.

a(n) = hypergeom([(1 - n)/2, 1 - n/2, -n], [1, 3/2], 1). - Vladimir Reshetnikov, Oct 15 2018

From Vladimir Reshetnikov, Oct 18 2018: (Start)

G.f.: 2^(1/3)*(6 - 8*x - 2^(1/3)*t^2)/(6*sqrt(x)*t), where t = (3*sqrt(12 - 39*x + 96*x^2) - (9 + 16*x)*sqrt(x))^(1/3).

D-finite with recurrence: 64*n*(n + 1)*(2*n + 1)*a(n) - 4*(n + 1)*(37*n^2 + 134*n + 120)*a(n + 1) + (n + 2)*(55*n^2 + 235*n + 240)*a(n + 2) - 2*(6*n + 21)*(n + 2)*(n + 3)*a(n + 3) = 0. (End)

Mathematica

Rest[CoefficientList[InverseSeries[Series[x/((1 + x) (1 - x^2)), {x, 0, 33}], x], x]]
Table[HypergeometricPFQ[{(1 - n)/2, 1 - n/2, -n}, {1, 3/2}, 1], {n, 1, 33}] (* Vladimir Reshetnikov, Oct 15 2018 *)

Crossrefs

Cf. A001057, A036765.

Sequence in context: A324184 A114691 A023639 * A331733 A301755 A302558

Adjacent sequences: A291531 A291532 A291533 * A291535 A291536 A291537

Keyword

sign

Author

Ilya Gutkovskiy, Aug 25 2017

Status

approved