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A008413 Coordination sequence for 5-dimensional cubic lattice.

%I A008413 #44 Mar 09 2023 09:46:44
%S A008413 1,10,50,170,450,1002,1970,3530,5890,9290,14002,20330,28610,39210,
%T A008413 52530,69002,89090,113290,142130,176170,216002,262250,315570,376650,
%U A008413 446210,525002,613810,713450,824770,948650,1086002,1237770,1404930
%N A008413 Coordination sequence for 5-dimensional cubic lattice.
%C A008413 If Y_i (i=1,2,3,4,5) are 2-blocks of a (n+5)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4,5). - _Milan Janjic_, Oct 28 2007
%H A008413 Seiichi Manyama, <a href="/A008413/b008413.txt">Table of n, a(n) for n = 0..10000</a>
%H A008413 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A008413 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A008413 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 A008413 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A008413 G.f.: ((1+x)/(1-x))^5.
%F A008413 a(n) = (4/3)*n^4 + (20/3)*n^2 + 2 for n > 0. - _Michael De Vlieger_, Oct 04 2016
%F A008413 n*a(n) = 10*a(n-1) + (n-2)*a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018
%F A008413 From _Shel Kaphan_, Mar 03 2023: (Start)
%F A008413 a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=5, for n>=1.
%F A008413 a(n) = A035599(n)*5/n, for n>0. (End)
%p A008413 4/3*n^4+20/3*n^2+2;
%t A008413 LinearRecurrence[{5,-10,10,-5,1},{1,10,50,170,450,1002},40] (* _Harvey P. Dale_, May 02 2016 *)
%t A008413 {1}~Join~Table[4/3 n^4 + 20/3 n^2 + 2, {n, 32}] (* or *)
%t A008413 CoefficientList[Series[((1 + x)/(1 - x))^5, {x, 0, 32}], x] (* _Michael De Vlieger_, Oct 04 2016 *)
%Y A008413 Cf. A035599.
%Y A008413 Row 5 of A035607, A266213.
%Y A008413 Column 5 of A113413, A119800, A122542.
%K A008413 nonn,easy
%O A008413 0,2
%A A008413 _N. J. A. Sloane_