A008417 Crystal ball sequence for 8-dimensional cubic lattice.

1, 17, 145, 833, 3649, 13073, 40081, 108545, 265729, 598417, 1256465, 2485825, 4673345, 8405905, 14546705, 24331777, 39490049, 62390545, 96220561, 145198913, 214828609, 312193553, 446304145, 628496897, 872893441, 1196924561, 1621925137, 2173806145, 2883810113, 3789356689, 4934985233

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

This is row/column 8 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - Shel Kaphan, Jan 06 2023

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).
• Index entries for crystal ball sequences
• Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Crossrefs

Cf. A001849, A099196.

Partial sums of A008416.

Cf. A240876.

Row/Column 8 of A008288.

Programs

Mathematica

CoefficientList[Series[-(z + 1)^8/(z - 1)^9, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,17,145,833,3649,13073,40081,108545,265729},40] (* Harvey P. Dale, May 26 2024 *)

Formula

G.f.: (1+x)^8/(1-x)^9.

First differences of A099196. - Alexander Adamchuk, May 23 2006

a(n) = (2*n^8 + 8*n^7 + 84*n^6 + 224*n^5 + 798*n^4 + 1232*n^3 + 1636*n^2 + 1056*n + 315)/315. - Alexander Adamchuk, May 23 2006

Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = log(2) - (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8). - Peter Bala, Mar 23 2024

Extensions

More terms from Alexander Adamchuk, May 23 2006