OFFSET
0,2
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
a(n) = S(n,6) = 7*n*(11*n^4 + 35*n^2 + 14)/10, with S(n,m) = Sum_{k=0..m} binomial(m,k)^2 * binomial(n-k+m-1, m-1), for n > 0, and a(0) = 1.
G.f.: (1+36*x+225*x^2+400*x^3+225*x^4+36*x^5+x^6)/(1-x)^6 = 1 + 42*x*(1+5*x+10*x^2+5*x^3+x^4)/(1-x)^6. - Colin Barker, Sep 26 2012
E.g.f.: 1 + (1/10)*x*(420 + 1890*x + 2170*x^2 + 770*x^3 + 77*x^4)*exp(x). - G. C. Greubel, May 26 2023
MAPLE
1, seq(7*n*(11*n^4+35*n^2+14)/10, n=1..40);
MATHEMATICA
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 42, 462, 2562, 9492, 27174, 65226}, 30] (* Jean-François Alcover, Jan 07 2019 *)
PROG
(Magma) [n eq 0 select 1 else 7*n*(11*n^4+35*n^2+14)/10: n in [0..50]]; // G. C. Greubel, May 26 2023
(SageMath) [7*n*(11*n^4 +35*n^2 +14)/10 +int(n==0) for n in range(51)] # G. C. Greubel, May 26 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved