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%I M4979 N2139 #59 Mar 23 2024 08:17:10
%S 1,15,113,575,2241,7183,19825,48639,108545,224143,433905,795455,
%T 1392065,2340495,3800305,5984767,9173505,13726991,20103025,28875327,
%U 40754369,56610575,77500017,104692735,139703809,184327311,240673265,311207743,398796225,506750351
%N Crystal ball sequence for 7-dimensional cubic lattice.
%C This is row/column 7 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - _Shel Kaphan_, Jan 06 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%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="/A001849/b001849.txt">Table of n, a(n) for n = 0..1000</a>
%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 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="https://www.jstor.org/stable/2029227">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_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F G.f.: (1+x)^7 /(1-x)^8.
%F a(n) = (8*n^7 + 28*n^6 + 224*n^5 + 490*n^4 + 1232*n^3 + 1372*n^2 + 1056*n + 315)/315. - _Johannes W. Meijer_, Jul 14 2013
%F Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = 319/420 - log(2) = (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7) - log(2). - _Peter Bala_, Mar 23 2024
%p A001849:=(z+1)**7/(z-1)**8; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t CoefficientList[Series[(z + 1)^7/(z - 1)^8, {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 19 2011 *)
%Y Cf. A001848, A001849.
%Y Cf. A240876.
%Y Row/column 7 of A008288.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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