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A001847 Crystal ball sequence for 5-dimensional cubic lattice.
(Formerly M4793 N2045)
10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

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).

LINKS

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

CoefficientList[Series[(z+1)^5/(z-1)^6, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

CROSSREFS

Cf. A005408, A001844, A001845, A001846, A013609.

Cf. A240876.

Sequence in context: A060884 A141935 A222408 * A089764 A023298 A106992

Adjacent sequences:  A001844 A001845 A001846 * A001848 A001849 A001850

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 20 16:25 EST 2018. Contains 299380 sequences. (Running on oeis4.)