OFFSET
0,3
COMMENTS
The first formula intersperses the terms with zeros, the second formula does not. - Colin Barker, Apr 02 2015
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).
FORMULA
G.f.: (1 +x^4 +x^6 +2*x^8 +x^10 +x^12 +x^16)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)), even powers only.
G.f.: (1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/ ((1-x)^4*(1+x)^2*(1+x^2)*(1+x+x^2)). - Colin Barker, Apr 02 2015
EXAMPLE
G.f. = 1 + x^2 + 3*x^4 + 5*x^6 + 10*x^8 + 14*x^10 + 23*x^12 + 31*x^14 + 46*x^16 + ...
MAPLE
S:= series((1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/mul(1-x^j, j=1..4), x, 65):
seq(coeff(S, x, j), j = 0..60); # G. C. Greubel, Feb 02 2020
MATHEMATICA
CoefficientList[Series[(1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), {x, 0, 60}], x] (* G. C. Greubel, Feb 02 2020 *)
PROG
(Magma) // Definition of group: F<al> := CyclotomicField(24); M := GeneralLinearGroup(4, F);
B1 := M![ -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1 ]; C1 := M![1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0 ];
C2 := M![0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0 ]; G := sub<M | B1, C1, C2 >;
(PARI) Vec((x^8+x^6+x^5+2*x^4+x^3+x^2+1) / ((x-1)^4*(x+1)^2*(x^2+1)*(x^2+x+1)) + O(x^60)) \\ Colin Barker, Apr 02 2015
(Sage)
def A078411_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) ).list()
A078411_list(60) # G. C. Greubel, Feb 02 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 27 2002
STATUS
approved