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Expansion of (1-x^7)/(1-x)^7.
3

%I #49 Sep 08 2022 08:44:35

%S 1,7,28,84,210,462,924,1715,2996,4977,7924,12166,18102,26208,37044,

%T 51261,69608,92939,122220,158536,203098,257250,322476,400407,492828,

%U 601685,729092,877338,1048894,1246420,1472772,1731009,2024400,2356431,2730812,3151484

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

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

%C Growth series of the affine Weyl group of type A6. - _Paul E. Gunnells_, Jan 06 2017

%D 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.

%D J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.

%H Colin Barker, <a href="/A008489/b008489.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Beck and S. Hosten, <a href="http://arxiv.org/abs/math/0508136">Cyclotomic polytopes and growth series of cyclotomic lattices</a>, arXiv:math/0508136 [math.CO], 2005-2006.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F Equals binomial transform of [1, 6, 15, 20, 15, 6, 1, -1, 1, -1, 1, ...] - _Gary W. Adamson_, Apr 29 2008

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

%F G.f.: (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)/(1-x)^6. - _Colin Barker_, Mar 04 2015

%F 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

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

%t CoefficientList[(1-x^7)/(1-x)^7 + O[x]^40, x] (* _Jean-François Alcover_, Jan 09 2019 *)

%o (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

%o (Magma) [1] cat [7*n*(84 +35*n^2 +n^4)/120: n in [1..40]]; // _G. C. Greubel_, Nov 07 2019

%o (Sage) [1]+[7*n*(84 +35*n^2 +n^4)/120 for n in (1..40)] # _G. C. Greubel_, Nov 07 2019

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_