A286899 Array read by antidiagonals: A(n, L) is the number of closed walks of length 2L along the edges of an n-cube based at a vertex, for n >= 1 and L >= 1.

1, 1, 2, 1, 8, 3, 1, 32, 21, 4, 1, 128, 183, 40, 5, 1, 512, 1641, 544, 65, 6, 1, 2048, 14763, 8320, 1205, 96, 7, 1, 8192, 132861, 131584, 26465, 2256, 133, 8, 1, 32768, 1195743, 2099200, 628805, 64896, 3787, 176, 9, 1, 131072, 10761681, 33562624, 15424865

Offset: 1

Author

Melvin Peralta, May 15 2017

Keywords

• nonn
• walk
• tabl
• easy

Examples

			A(2, 2) = 8 because at each vertex of a 2-cube (i.e., a square), there are 8 closed walks of length 2(2) = 4.
A(1, k) = 1 because at the vertex of a 1-cube, there is 1 closed walk of any length 2*k.
Array A(n, L) begins:
   1         1         1         1         1         1 ...
   2         8        32       128       512      2048 ...
   3        21       183      1641     14763    132861 ...
   4        40       544      8320    131584   2099200 ...
   5        65      1205     26465    628805  15424865 ...
   6        96      2256     64896   2086656  71172096 ...
   7       133      3787    134953   5501167 243147373 ...
		

References

• R. P. Stanley, Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer, 2013.

Links

• Melvin Peralta, Table of n, a(n) for n = 1..120

Crossrefs

Cf. A054879, A092812, A121822.

Programs

Maple

A286899 := proc(n,L)
    add(binomial(n,i)*(n-2*i)^L, i=0..n) ;
    %/2^n ;
end proc:
for n from 1 to 7 do
    for L from 2 to 12 by 2 do
        printf("%9d ",A286899(n,L)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 22 2017

Mathematica

f[n_, l_] := 1/2^n*Sum[Binomial[n, i]*(n - 2 i)^l, {i, 0, n}];
Table[f[n - l + 1, 2 l], {n, 1, 15}, {l, n, 1, -1}] // Flatten

Formula

A(n, L) = (1/2^n)*Sum_{i=0..n} binomial(n, i)*(n - 2*i)^(2*L). (Corrected by Peter Luschny, Jul 07 2019.)