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%I A008387 #40 Dec 13 2023 10:54:40
%S A008387 1,42,462,2562,9492,27174,65226,137886,264936,472626,794598,1272810,
%T A008387 1958460,2912910,4208610,5930022,8174544,11053434,14692734,19234194,
%U A008387 24836196,31674678,39944058,49858158,61651128,75578370,91917462,110969082
%N A008387 Coordination sequence for A_6 lattice.
%H A008387 T. D. Noe, <a href="/A008387/b008387.txt">Table of n, a(n) for n = 0..1000</a>
%H A008387 M. Baake and U. Grimm, <a href="https://arxiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
%H A008387 R. Bacher, P. de la Harpe and B. Venkov, <a href="https://doi.org/10.1016/S0764-4442(97)83542-2">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H A008387 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 A008387 M. O'Keeffe, <a href="http://dx.doi.org/10.1524/zkri.1995.210.12.905">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908.
%H A008387 M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
%H A008387 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A008387 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.
%F A008387 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
%F A008387 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
%p A008387 1, seq(7*n*(11*n^4+35*n^2+14)/10, n=1..40);
%t A008387 LinearRecurrence[{6,-15,20,-15,6,-1}, {1,42,462,2562,9492,27174,65226}, 30] (* _Jean-François Alcover_, Jan 07 2019 *)
%o A008387 (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
%o A008387 (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
%Y A008387 Row 6 of A103881.
%K A008387 nonn,easy
%O A008387 0,2
%A A008387 _N. J. A. Sloane_ and _J. H. Conway_

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