A049018 Expansion of 1/((1+x)^7 - x^7).

1, -7, 28, -84, 210, -462, 924, -1715, 2989, -4900, 7448, -9996, 9996, 0, -38759, 149205, -422576, 1041348, -2350922, 4970070, -9940140, 18874261, -33957343, 57374296, -89125120, 120875944, -120875944, 0, 459957169, -1749692735, 4904887652

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, signature (-7, -21, -35, -35, -21, -7).

Crossrefs

Column 7 of A307047.

Cf. A049017.

Programs

Magma

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( 1/((1+x)^7 - x^7) )); // G. C. Greubel, Mar 17 2019

Mathematica

LinearRecurrence[{-7,-21,-35,-35,-21,-7},{1,-7,28,-84,210,-462}, 35] (* Ray Chandler, Sep 23 2015 *)

PARI

Vec(1/((1+x)^7-x^7)+O(x^35)) \\ Charles R Greathouse IV, Sep 27 2012

PARI

{a(n) = (-1)^n*sum(k=0, n\7, (-1)^k*binomial(n+6, 7*k+6))} \\ Seiichi Manyama, Mar 21 2019

Sage

(1/((1+x)^7 - x^7)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Mar 17 2019

Formula

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