OFFSET
0,2
COMMENTS
Related to the 4-dimensional cyclotomic lattice Z[zeta_5] (or A_4^{*}).
Growth series of the affine Weyl group of type A4. - 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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
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
Equals binomial transform of [1, 4, 6, 4, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 29 2008
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
For n >= 1, a(n) = (5*n^3 + 25*n)/6. - Christopher Hohl, Dec 30 2018
E.g.f.: 1 + x*(30 + 15*x + 5*x^2)*exp(x)/6. - G. C. Greubel, Nov 07 2019
MAPLE
1, seq(5*n*(n^2 +5)/6, n=1..50); # G. C. Greubel, Nov 07 2019
MATHEMATICA
CoefficientList[Series[(1-x^5)/(1-x)^5, {x, 0, 50}], x] (* Stefano Spezia, Dec 30 2018 *)
PROG
(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
(Magma) [1] cat [5*n*(n^2 +5)/6: n in [1..50]]; // G. C. Greubel, Nov 07 2019
(Sage) [1]+[5*n*(n^2 +5)/6 for n in (1..50)] # G. C. Greubel, Nov 07 2019
(GAP) concatenation([1], List([1..50], n-> 5*n*(n^2 +5)/6)); # G. C. Greubel, Nov 07 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved