A239286 Expansion of (x + 1)*(3*x^2 + 2*x + 1)/(x^2 + x + 1)^2.

1, 1, 0, -2, 1, 3, -5, 1, 6, -8, 1, 9, -11, 1, 12, -14, 1, 15, -17, 1, 18, -20, 1, 21, -23, 1, 24, -26, 1, 27, -29, 1, 30, -32, 1, 33, -35, 1, 36, -38, 1, 39, -41, 1, 42, -44, 1, 45, -47, 1, 48, -50, 1, 51, -53, 1, 54, -56, 1, 57, -59, 1, 60, -62, 1, 63, -65

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Index entries for linear recurrences with constant coefficients, signature (-2,-3,-2,-1).

Formula

a(n) = Sum_{k=ceiling((n-1)/2)..n} (-1)^(k+1)*C(k+1,n-k)*(k*n-1)/(k+1).

a(3n+1) = 1, a(3n) = 1 - 3n, a(3n+2) = 3n. - Ralf Stephan, Mar 17 2014

Mathematica

CoefficientList[Series[(x+1)*(3*x^2+2*x+1)/(x^2+x+1)^2, {x, 0, 50}], x] (* G. C. Greubel, May 26 2018 *)
(* Alternative: *)
LinearRecurrence[{-2, -3, -2, -1}, {1, 1, 0, -2}, 30] (* G. C. Greubel, May 26 2018 *)

Prog

(Maxima) a(n):=sum(((-1)^(k+1)*binomial(k+1, n-k)*(k*n-1))/(k+1), k, ceiling((n-1)/2), n);
(PARI) my(x='x+O('x^30)); Vec((x+1)*(3*x^2+2*x+1)/(x^2+x+1)^2) \\ G. C. Greubel, May 26 2018

Crossrefs

Sequence in context: A220562 A215564 A189449 * A109533 A062705 A059234

Adjacent sequences: A239283 A239284 A239285 * A239287 A239288 A239289

Keyword

sign,easy,changed

Author

Vladimir Kruchinin, Mar 14 2014

Status

approved