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%I M4793 N2045 #80 Mar 23 2024 08:19:04
%S 1,11,61,231,681,1683,3653,7183,13073,22363,36365,56695,85305,124515,
%T 177045,246047,335137,448427,590557,766727,982729,1244979,1560549,
%U 1937199,2383409,2908411,3522221,4235671,5060441,6009091,7095093,8332863,9737793,11326283
%N Crystal ball sequence for 5-dimensional cubic lattice.
%C Number of nodes degree 10 in virtual, optimal chordal graphs of diameter d(G)=n - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 07 2002
%C If Y_i (i=1,2,3,4,5) are 2-blocks of a (n+5)-set X then a(n-5) is the number of 10-subsets of X intersecting each Y_i (i=1,2,3,4,5). - _Milan Janjic_, Oct 28 2007
%C Equals binomial transform of [1, 10, 40, 80, 80, 32, 0, 0, 0, ...] where (1, 10, 40, 80, 80, 32) = row 5 of the Chebyshev triangle A013609. - _Gary W. Adamson_, Jul 19 2008
%C a(n) is the number of points in Z^5 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 5 from any given point. - _Shel Kaphan_, Jan 02 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 231.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001847/b001847.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, <a href="http://www.cecm.sfu.ca/~choi/paper/lrh.pdf">A local Riemann hypothesis, I</a> pages 16 and 17
%H 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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>. [Broken link; <a href="https://web.archive.org/web/20110204173116/http://www.pmfbl.org:80/janjic/">WayBackMachine archive</a>]
%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
%H G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H R. G. Stanton and D. D. Cowan, <a href="http://dx.doi.org/10.1137/1012049">Note on a "square" functional equation</a>, SIAM Rev., 12 (1970), 277-279.
%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F G.f.: (1+x)^5 /(1-x)^6.
%F a(n) = (4*n^5+10*n^4+40*n^3+50*n^2+46*n+15)/15. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 07 2002
%F a(n) = Sum_{k=0..min(5,n)} 2^k * binomial(5,k)* binomial(n,k). See Bump et al. - _Tom Copeland_, Sep 05 2014
%F E.g.f.: exp(x)*(15 + 150*x + 300*x^2 + 200*x^3 + 50*x^4 + 4*x^5)/15. - _Stefano Spezia_, Mar 17 2024
%F Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = 47/60 - log(2) = (1 - 1/2 + 1/3 - 1/4 + 1/5) - log(2). - _Peter Bala_, Mar 23 2024
%e a(5)=1683, (4*5^5 + 10*5^4 + 40*5^3 + 50*5^2 + 46*5 + 15)/15 = (12500 + 6250 + 5000 + 230 + 15)/15 = 25245/15 = 1683.
%p for n from 1 to k do eval((4*n^5+10*n^4+40*n^3+50*n^2+46*n+15)/15) od;
%p A001847:=(z+1)**5/(z-1)**6; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t CoefficientList[Series[(z+1)^5/(z-1)^6, {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 19 2011 *)
%Y Cf. A005408, A001844, A001845, A001846, A013609.
%Y Cf. A240876.
%Y Row/column 5 of A008288.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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