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A001847
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Crystal ball sequence for 5-dimensional cubic lattice.
(Formerly M4793 N2045)
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10
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1, 11, 61, 231, 681, 1683, 3653, 7183, 13073, 22363, 36365, 56695, 85305, 124515, 177045, 246047, 335137, 448427, 590557, 766727, 982729, 1244979, 1560549, 1937199, 2383409, 2908411, 3522221, 4235671, 5060441, 6009091, 7095093, 8332863, 9737793, 11326283
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OFFSET
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0,2
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COMMENTS
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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
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
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
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 231.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, A local Riemann hypothesis, I pages 16 and 17
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Milan Janjic, Two Enumerative Functions
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
Index entries for crystal ball sequences
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FORMULA
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G.f.: (1+x)^5 /(1-x)^6.
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
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
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EXAMPLE
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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.
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MAPLE
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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;
A001847:=(z+1)**5/(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation
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MATHEMATICA
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CoefficientList[Series[(z+1)^5/(z-1)^6, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
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CROSSREFS
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Cf. A005408, A001844, A001845, A001846, A013609.
Cf. A240876.
Sequence in context: A060884 A141935 A222408 * A089764 A023298 A320145
Adjacent sequences: A001844 A001845 A001846 * A001848 A001849 A001850
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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