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Expansion of (1-x^5) / (1-x)^5.
3

%I #61 Sep 08 2022 08:44:35

%S 1,5,15,35,70,125,205,315,460,645,875,1155,1490,1885,2345,2875,3480,

%T 4165,4935,5795,6750,7805,8965,10235,11620,13125,14755,16515,18410,

%U 20445,22625,24955,27440,30085,32895,35875,39030,42365,45885,49595,53500,57605,61915

%N Expansion of (1-x^5) / (1-x)^5.

%C Related to the 4-dimensional cyclotomic lattice Z[zeta_5] (or A_4^{*}).

%C Growth series of the affine Weyl group of type A4. - _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="/A008487/b008487.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 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) is the sum of 5 consecutive tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6 = A000292(n) for n>0, a(0) = 1. a(n) = A000292(n-4) + A000292(n-3) + A000292(n-2) + A000292(n-1) + A000292(n) for n>0, a(0) = 1. - _Alexander Adamchuk_, May 20 2006

%F Equals binomial transform of [1, 4, 6, 4, 1, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Apr 29 2008

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. - _Colin Barker_, Jan 06 2017

%F For n >= 1, a(n) = (5*n^3 + 25*n)/6. - _Christopher Hohl_, Dec 30 2018

%F E.g.f.: 1 + x*(30 + 15*x + 5*x^2)*exp(x)/6. - _G. C. Greubel_, Nov 07 2019

%p 1, seq(5*n*(n^2 +5)/6, n=1..50); # _G. C. Greubel_, Nov 07 2019

%t CoefficientList[Series[(1-x^5)/(1-x)^5, {x, 0, 50}], x] (* _Stefano Spezia_, Dec 30 2018 *)

%o (PARI) Vec((1-x^5) / (1-x)^5+O(x^50)) \\ _Charles R Greathouse IV_, Sep 26 2012; corrected by _Colin Barker_, Jan 06 2017

%o (Magma) [1] cat [5*n*(n^2 +5)/6: n in [1..50]]; // _G. C. Greubel_, Nov 07 2019

%o (Sage) [1]+[5*n*(n^2 +5)/6 for n in (1..50)] # _G. C. Greubel_, Nov 07 2019

%o (GAP) concatenation([1], List([1..50], n-> 5*n*(n^2 +5)/6)); # _G. C. Greubel_, Nov 07 2019

%Y Cf. A000292, A008498, A008531, A222408.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_