A001849 Crystal ball sequence for 7-dimensional cubic lattice.

1, 15, 113, 575, 2241, 7183, 19825, 48639, 108545, 224143, 433905, 795455, 1392065, 2340495, 3800305, 5984767, 9173505, 13726991, 20103025, 28875327, 40754369, 56610575, 77500017, 104692735, 139703809, 184327311, 240673265, 311207743, 398796225, 506750351

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
• 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
• 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; 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
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A001848, A001849.

Cf. A240876.

Row/column 7 of A008288.

Programs

Maple

A001849:=(z+1)**7/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[Series[(z + 1)^7/(z - 1)^8, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

Formula

G.f.: (1+x)^7 /(1-x)^8.

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

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