A008489 Expansion of (1-x^7)/(1-x)^7.

1, 7, 28, 84, 210, 462, 924, 1715, 2996, 4977, 7924, 12166, 18102, 26208, 37044, 51261, 69608, 92939, 122220, 158536, 203098, 257250, 322476, 400407, 492828, 601685, 729092, 877338, 1048894, 1246420, 1472772, 1731009, 2024400, 2356431, 2730812, 3151484

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Coordination sequence for 6-dimensional cyclotomic lattice Z[zeta_7].

Growth series of the affine Weyl group of type A6. - Paul E. Gunnells, Jan 06 2017

References

• R. Bott, The geometry and the representation theory of compact Lie groups, in: Representation Theory of Lie Groups, in: London Math. Soc. Lecture Note Ser., vol. 34, Cambridge University Press, Cambridge, 1979, pp. 65-90.
• J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Programs

GAP

Concatenation([1], List([1..40], n-> 7*n*(84 +35*n^2 +n^4)/120)); # G. C. Greubel, Nov 07 2019

Magma

[1] cat [7*n*(84 +35*n^2 +n^4)/120: n in [1..40]]; // G. C. Greubel, Nov 07 2019

Maple

1, seq(7*n*(84 +35*n^2 +n^4)/120, n=1..40); # G. C. Greubel, Nov 07 2019

Mathematica

CoefficientList[(1-x^7)/(1-x)^7 + O[x]^40, x] (* Jean-François Alcover, Jan 09 2019 *)

PARI

Vec((x^6+x^5+x^4+x^3+x^2+x+1)/(x-1)^6 + O(x^40)) \\ Colin Barker, Mar 04 2015

Sage

[1]+[7*n*(84 +35*n^2 +n^4)/120 for n in (1..40)] # G. C. Greubel, Nov 07 2019

Formula

Equals binomial transform of [1, 6, 15, 20, 15, 6, 1, -1, 1, -1, 1, ...] - Gary W. Adamson, Apr 29 2008

a(n) = 7*n*(84 + 35*n^2 + n^4)/120, n>0. - R. J. Mathar, Mar 17 2011

G.f.: (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)/(1-x)^6. - Colin Barker, Mar 04 2015

E.g.f.: 1 + x*(840 + 840*x + 420*x^2 + 70*x^3 + 7*x^4)*exp(x)/120. - G. C. Greubel, Nov 07 2019