A192080 Expansion of 1/((1-x)^6 - x^6).

1, 6, 21, 56, 126, 252, 463, 804, 1365, 2366, 4368, 8736, 18565, 40410, 87381, 184604, 379050, 758100, 1486675, 2884776, 5592405, 10919090, 21572460, 43144920, 87087001, 176565486, 357913941, 723002336, 1453179126, 2906358252

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Formula

a(n) = abs(A006090(n)) = (-1)^n * A006090(n).

G.f.: 1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)).

From G. C. Greubel, Apr 11 2023: (Start)

a(n) = (2^(n+5) + A010892(n) - 2*A010892(n-1) - 27*(A057083(n) - 2*A057083(n-1)))/6.

a(n) = (2^(n+5) + A057079(n+2) - 27*A057681(n+1))/6. (End)

Mathematica

CoefficientList[Series[1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2012 *)
LinearRecurrence[{6, -15, 20, -15, 6}, {1, 6, 21, 56, 126}, 30] (* Harvey P. Dale, Feb 22 2017 *)

Prog

(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))));
(Maxima) makelist(coeff(taylor(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), x, 0, n), x, n), n, 0, 29);
(PARI) Vec(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))+O(x^99)) \\ Charles R Greathouse IV, Jun 23 2011
(SageMath)
def A192080_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^6-x^6) ).list()
A192080_list(51) # G. C. Greubel, Apr 11 2023

Crossrefs

Cf. A006090, A049016, A049017, A057079, A057083, A057681.

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), this sequence (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

Sequence in context: A140228 A264926 A006090 * A290993 A275936 A019500

Adjacent sequences: A192077 A192078 A192079 * A192081 A192082 A192083

Keyword

nonn,easy

Author

Bruno Berselli, Jun 23 2011

Status

approved