A228706 Expansion of (1 - 3*x + 5*x^2 - 3*x^3 + x^4)/((1-x)^4*(1+x^2)^2).

1, 1, 1, 5, 11, 14, 18, 30, 45, 55, 67, 91, 119, 140, 164, 204, 249, 285, 325, 385, 451, 506, 566, 650, 741, 819, 903, 1015, 1135, 1240, 1352, 1496, 1649, 1785, 1929, 2109, 2299, 2470, 2650, 2870, 3101, 3311, 3531, 3795, 4071, 4324, 4588, 4900, 5225, 5525

Offset: 0

Author

N. J. A. Sloane, Sep 06 2013

Keywords

• nonn
• easy

Comments

A159914 and A228705 both satisfy the same recurrence relation, and both count (n-3)-element subsets of {1..n} having even resp. odd sum. Is there a similar subset-counting interpretation for this sequence? - M. F. Hasler, Jun 22 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• E. Kirkman, J. Kuzmanovich and J. J. Zhang, Invariants of (-1)-Skew Polynomial Rings under Permutation Representations, arXiv preprint arXiv:1305.3973 [math.RA], 2013. See Example 5.6.
• Index entries for linear recurrences with constant coefficients, signature (4,-8,12,-14,12,-8,4,-1).

Crossrefs

Cf. A159914, A228705.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-3*x+5*x^2-3*x^3+x^4)/((1-x)^4*(1+x^2)^2)); // Vincenzo Librandi, Sep 07 2013

Mathematica

CoefficientList[Series[(1 - 3 x + 5 x^2 - 3 x^3 + x^4) / ((1 - x)^4 (1 + x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 07 2013 *)

PARI

Vec((1-3*x+5*x^2-3*x^3+x^4)/((1-x)^4*(1+x^2)^2)+O(x^99)) \\ M. F. Hasler, Jun 22 2018

Formula

a(n) = (n+2)*(2*(n+1)*(n+3)+9*(1+(-1)^n)*i^(n*(n+1)))/48, where i=sqrt(-1). [Bruno Berselli, Sep 07 2013]