A228705 Expansion of (1-2*x+4*x^2-2*x^3+x^4)/((1-x)^4*(1+x^2)^2).

1, 2, 4, 10, 19, 28, 40, 60, 85, 110, 140, 182, 231, 280, 336, 408, 489, 570, 660, 770, 891, 1012, 1144, 1300, 1469, 1638, 1820, 2030, 2255, 2480, 2720, 2992, 3281, 3570, 3876, 4218, 4579, 4940, 5320, 5740, 6181, 6622, 7084, 7590, 8119, 8648, 9200, 9800, 10425

Offset: 0

Author

N. J. A. Sloane, Sep 06 2013

Keywords

• nonn
• easy

Comments

Number of n-element subsets of [n+3] having an even sum. a(3) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}. - Alois P. Heinz, Feb 04 2017

A159914, which is half the number of (n-3)-element subsets of {1..n} having an odd sum, satisfies the same recurrence relation. However, a simple relation between a(n) and A159914(n) is not obvious. - 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, 2013. See Example 5.6.
• Index entries for linear recurrences with constant coefficients, signature (4, -8, 12, -14, 12, -8, 4, -1).

Crossrefs

Third lower diagonal of A282011.

Programs

Magma

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

Mathematica

CoefficientList[Series[(1 - 2 x + 4 x^2 - 2 x^3 + x^4) / ((1 - x)^4 (1 + x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 07 2013 *)
LinearRecurrence[{4,-8,12,-14,12,-8,4,-1},{1,2,4,10,19,28,40,60},50] (* Harvey P. Dale, Apr 10 2014 *)

Formula

a(n) = (n+2)*(2*(n+1)*(n+3)+3*(1+(-1)^n)*i^(n*(n+1)))/24, where i=sqrt(-1). [Bruno Berselli, Sep 07 2013]

a(0)=1, a(1)=2, a(2)=4, a(3)=10, a(4)=19, a(5)=28, a(6)=40, a(7)=60, a(n)=4*a(n-1)-8*a(n-2)+12*a(n-3)-14*a(n-4)+12*a(n-5)-8*a(n-6)+ 4*a(n-7)- a(n-8). - Harvey P. Dale, Apr 10 2014