A008413 Coordination sequence for 5-dimensional cubic lattice.

1, 10, 50, 170, 450, 1002, 1970, 3530, 5890, 9290, 14002, 20330, 28610, 39210, 52530, 69002, 89090, 113290, 142130, 176170, 216002, 262250, 315570, 376650, 446210, 525002, 613810, 713450, 824770, 948650, 1086002, 1237770, 1404930

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If Y_i (i=1,2,3,4,5) are 2-blocks of a (n+5)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4,5). - Milan Janjic, Oct 28 2007

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Milan Janjic, Two Enumerative Functions
• Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
• 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 linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A035599.

Row 5 of A035607, A266213.

Column 5 of A113413, A119800, A122542.

Programs

Maple

4/3*n^4+20/3*n^2+2;

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{1,10,50,170,450,1002},40] (* Harvey P. Dale, May 02 2016 *)
{1}~Join~Table[4/3 n^4 + 20/3 n^2 + 2, {n, 32}] (* or *)
CoefficientList[Series[((1 + x)/(1 - x))^5, {x, 0, 32}], x] (* Michael De Vlieger, Oct 04 2016 *)

Formula

G.f.: ((1+x)/(1-x))^5.

a(n) = (4/3)*n^4 + (20/3)*n^2 + 2 for n > 0. - Michael De Vlieger, Oct 04 2016

n*a(n) = 10*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

From Shel Kaphan, Mar 03 2023: (Start)

a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=5, for n>=1.

a(n) = A035599(n)*5/n, for n>0. (End)