A008387 Coordination sequence for A_6 lattice.
1, 42, 462, 2562, 9492, 27174, 65226, 137886, 264936, 472626, 794598, 1272810, 1958460, 2912910, 4208610, 5930022, 8174544, 11053434, 14692734, 19234194, 24836196, 31674678, 39944058, 49858158, 61651128, 75578370, 91917462, 110969082
Offset: 0
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).
Crossrefs
Row 6 of A103881.
Programs
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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
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Maple
1, seq(7*n*(11*n^4+35*n^2+14)/10, n=1..40);
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Mathematica
LinearRecurrence[{6,-15,20,-15,6,-1}, {1,42,462,2562,9492,27174,65226}, 30] (* Jean-François Alcover, Jan 07 2019 *) -
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
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