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Expansion of 1/((1+x)^7 - x^7).
5

%I #26 Sep 08 2022 08:44:58

%S 1,-7,28,-84,210,-462,924,-1715,2989,-4900,7448,-9996,9996,0,-38759,

%T 149205,-422576,1041348,-2350922,4970070,-9940140,18874261,-33957343,

%U 57374296,-89125120,120875944,-120875944,0,459957169,-1749692735,4904887652

%N Expansion of 1/((1+x)^7 - x^7).

%C It appears that the (unsigned) sequence is identical to its 7th-order absolute difference. - _John W. Layman_, Oct 02 2003

%H Seiichi Manyama, <a href="/A049018/b049018.txt">Table of n, a(n) for n = 0..3000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (-7, -21, -35, -35, -21, -7).

%F a(n) = (-1)^n * Sum_{k=0..floor(n/7)} (-1)^k * binomial(n+6,7*k+6). - _Seiichi Manyama_, Mar 21 2019

%t LinearRecurrence[{-7,-21,-35,-35,-21,-7},{1,-7,28,-84,210,-462}, 35] (* _Ray Chandler_, Sep 23 2015 *)

%o (PARI) Vec(1/((1+x)^7-x^7)+O(x^35)) \\ _Charles R Greathouse IV_, Sep 27 2012

%o (PARI) {a(n) = (-1)^n*sum(k=0, n\7, (-1)^k*binomial(n+6, 7*k+6))} \\ _Seiichi Manyama_, Mar 21 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 35); Coefficients(R!( 1/((1+x)^7 - x^7) )); // _G. C. Greubel_, Mar 17 2019

%o (Sage) (1/((1+x)^7 - x^7)).series(x, 35).coefficients(x, sparse=False) # _G. C. Greubel_, Mar 17 2019

%Y Column 7 of A307047.

%Y Cf. A049017.

%K sign,easy

%O 0,2

%A _N. J. A. Sloane_