OFFSET
1,3
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
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.)
EXAMPLE
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 ...
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
CROSSREFS
KEYWORD
AUTHOR
Melvin Peralta, May 15 2017
STATUS
approved