A001849 Crystal ball sequence for 7-dimensional cubic lattice. (Formerly M4979 N2139)

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

0,2

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).

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

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 *)

Crossrefs

Cf. A001848, A001849.

Cf. A240876.

Row/column 7 of A008288.

Sequence in context: A295384 A110822 A222410 * A115150 A115138 A233326

Adjacent sequences: A001846 A001847 A001848 * A001850 A001851 A001852

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved