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A001848 Crystal ball sequence for 6-dimensional cubic lattice.
(Formerly M4904 N2102)
7
1, 13, 85, 377, 1289, 3653, 8989, 19825, 40081, 75517, 134245, 227305, 369305, 579125, 880685, 1303777, 1884961, 2668525, 3707509, 5064793, 6814249, 9041957, 11847485, 15345233, 19665841, 24957661, 31388293, 39146185, 48442297, 59511829, 72616013, 88043969 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of nodes of degree 12 in virtual, optimal chordal graphs of diameter d(G)=n - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002

Equals binomial transform of [1, 12, 60, 160, 240, 192, 64, 0, 0, 0,...] where (1, 12, 60, 160, 240, 192, 64) = row 6 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).

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

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

FORMULA

G.f.: (1+x)^6 /(1-x)^7.

a(n) = 4/45*n^6+4/15*n^5+14/9*n^4+8/3*n^3+196/45*n^2+46/15*n+1. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002

a(n)= sum(k=0..min(6,n), 2^k * binomial(6,k)* binomial(n,k) ). See Bump et al. - Tom Copeland, Sep 05 2014

MAPLE

for n from 1 to k do eval(4/45*n^6+4/15*n^5+14/9*n^4+8/3*n^3+196/45*n^2+46/15*n+1); od;

A001848:=-(z+1)**6/(z-1)**7; # [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

CoefficientList[Series[-(z + 1)^6/(z - 1)^7, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

CROSSREFS

Cf. A001847, A013609.

Cf. A240876.

Sequence in context: A297207 A222491 A010025 * A055843 A296647 A233325

Adjacent sequences:  A001845 A001846 A001847 * A001849 A001850 A001851

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified February 24 18:55 EST 2021. Contains 341584 sequences. (Running on oeis4.)