A229955 Triangular array read by rows: 3 dimensional analog of A227997.

8, 152, 64, 5056, 2432, 512, 205720, 104000, 29184, 4096, 9305152, 4828544, 1525248, 311296, 32768, 449404224, 236984448, 79898624, 19226624, 3112960, 262144, 22695553536, 12099474432, 4251479040, 1123909632, 221839360, 29884416, 2097152, 1183891745688, 636162156096, 230017430016, 64636047360, 14330265600, 2413559808, 278921216, 16777216

Offset: 1

Author

Geoffrey Critzer, Oct 04 2013

Keywords

• nonn
• tabl
• walk

Comments

T(n,k) is the number of walks on the 3 dimensional grid that start and end at the origin using 2n steps and having exactly k primitive loops. The steps are in the eight directions: (1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1), (-1,1,1), (-1,1,-1), (-1,-1,1), (-1,-1,-1). A primitive loop is a walk that starts and ends on the origin but does not otherwise touch the origin.

Column 1 is A094059.

Row sums are A002897.

Examples

			8,
152, 64,
5056, 2432, 512,
205720, 104000, 29184, 4096,
9305152, 4828544, 1525248, 311296, 32768,
449404224, 236984448, 79898624, 19226624, 3112960, 262144
		

Programs

Mathematica

nn=6;a=Sum[Binomial[2n,n]^3x^n,{n,0,nn}];Map[Select[#,#>0&]&,Drop[CoefficientList[Series[1/(1-y(1-1/a)),{x,0,nn}],{x,y}],1]]//Grid

Formula

G.f.: 1/( 1 - y*(1 - 1/A(x)) ) where A(x) is the o.g.f. for A002897.

Generally for such walks in N dimensions: 1/( 1 - y*(1 - 1/B(x)) ) where B(x) = Sum_{n>=0} binomial(2n,n)^N*x^n.