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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.

%I A286899 #29 Jul 08 2019 01:32:01
%S A286899 1,1,2,1,8,3,1,32,21,4,1,128,183,40,5,1,512,1641,544,65,6,1,2048,
%T A286899 14763,8320,1205,96,7,1,8192,132861,131584,26465,2256,133,8,1,32768,
%U A286899 1195743,2099200,628805,64896,3787,176,9,1,131072,10761681,33562624,15424865
%N 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.
%D A286899 R. P. Stanley, Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer, 2013.
%H A286899 Melvin Peralta, <a href="/A286899/b286899.txt">Table of n, a(n) for n = 1..120</a>
%F A286899 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.)
%e A286899 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.
%e A286899 A(1, k) = 1 because at the vertex of a 1-cube, there is 1 closed walk of any length 2*k.
%e A286899 Array A(n, L) begins:
%e A286899    1         1         1         1         1         1 ...
%e A286899    2         8        32       128       512      2048 ...
%e A286899    3        21       183      1641     14763    132861 ...
%e A286899    4        40       544      8320    131584   2099200 ...
%e A286899    5        65      1205     26465    628805  15424865 ...
%e A286899    6        96      2256     64896   2086656  71172096 ...
%e A286899    7       133      3787    134953   5501167 243147373 ...
%p A286899 A286899 := proc(n,L)
%p A286899     add(binomial(n,i)*(n-2*i)^L, i=0..n) ;
%p A286899     %/2^n ;
%p A286899 end proc:
%p A286899 for n from 1 to 7 do
%p A286899     for L from 2 to 12 by 2 do
%p A286899         printf("%9d ",A286899(n,L)) ;
%p A286899     end do:
%p A286899     printf("\n") ;
%p A286899 end do: # _R. J. Mathar_, May 22 2017
%t A286899 f[n_, l_] := 1/2^n*Sum[Binomial[n, i]*(n - 2 i)^l, {i, 0, n}];
%t A286899 Table[f[n - l + 1, 2 l], {n, 1, 15}, {l, n, 1, -1}] // Flatten
%Y A286899 Cf. A054879, A092812, A121822.
%K A286899 nonn,walk,tabl,easy
%O A286899 1,3
%A A286899 _Melvin Peralta_, May 15 2017