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A209349 Number A(n,k) of initially rising meander words, where each letter of the cyclic k-ary alphabet occurs n times; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A209349 #20 Jan 21 2015 04:44:45
%S A209349 1,1,1,1,1,1,1,1,0,1,1,1,1,0,1,1,1,5,1,0,1,1,1,9,29,1,0,1,1,1,11,100,
%T A209349 182,1,0,1,1,1,16,182,1225,1198,1,0,1,1,1,19,484,3542,15876,8142,1,0,
%U A209349 1,1,1,25,902,17956,76258,213444,56620,1,0,1
%N A209349 Number A(n,k) of initially rising meander words, where each letter of the cyclic k-ary alphabet occurs n times; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A209349 In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too.  The words are not considered cyclic here.
%C A209349 A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.
%C A209349 A(n,k) is also the number of (n*k-1)-step walks on k-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by 1 or are in the set {1,k}.
%H A209349 Alois P. Heinz, <a href="/A209349/b209349.txt">Antidiagonals n = 0..15, flattened</a>
%e A209349 A(0,0) = A(0,k) = A(n,0) = 1: the empty word.
%e A209349 A(1,1) = 1 = |{a}|.
%e A209349 A(2,1) = 0 = |{ }|.
%e A209349 A(2,2) = 1 = |{abab}|.
%e A209349 A(2,3) = 5 = |{abacbc, abcabc, abcacb, abcbac, abcbca}|.
%e A209349 A(1,4) = 1 = |{abcd}|.
%e A209349 A(2,4) = 9 = |{ababcdcd, abadcbcd, abadcdcb, abcbadcd, abcbcdad, abcdabcd, abcdadcb, abcdcbad, abcdcdab}|.
%e A209349 Square array A(n,k) begins:
%e A209349 1,  1,  1,    1,      1,       1,        1, ...
%e A209349 1,  1,  1,    1,      1,       1,        1, ...
%e A209349 1,  0,  1,    5,      9,      11,       16, ...
%e A209349 1,  0,  1,   29,    100,     182,      484, ...
%e A209349 1,  0,  1,  182,   1225,    3542,    17956, ...
%e A209349 1,  0,  1, 1198,  15876,   76258,   749956, ...
%e A209349 1,  0,  1, 8142, 213444, 1753522, 33779344, ...
%p A209349 b:= proc() option remember; local n; n:= nargs;
%p A209349      `if`({args}={0}, 1,
%p A209349        `if`(args[2]>0, b(args[2]-1, args[i]$i=3..n, args[1]), 0)+
%p A209349        `if`(n>2 and args[n]>0, b(args[n]-1, args[i]$i=1..n-1), 0))
%p A209349     end:
%p A209349 A:= (n, k)-> `if`(n<2, 1, `if`(k<2, 1-k, b((n-1)$2, n$(k-2)))):
%p A209349 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A209349 b[args_List] := b[args] = Module[{n = Length[args]}, If[Union[args] == {0}, 1, If[args[[2]] > 0, b[Join[{args[[2]] - 1}, args[[3 ;; n]], { args[[1]]}]], 0] + If[n > 2 && args[[n]] > 0, b[Join[{args[[n]] - 1}, args[[1 ;; n - 1]]]], 0]]]; A[n_, k_] := If[n < 2, 1, If[k < 2, 1 - k, b[Join[{n - 1, n - 1}, Array[n&, k - 2]]]]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 21 2015, after _Alois P. Heinz_ *)
%Y A209349 Rows n=0+1, 2-3 give: A000012, A209350, A240954.
%Y A209349 Columns k=0+2, 3-7 give: A000012, A190917 = A110706/6, A060150 = A088218^2, A209351, A209352, A209353.
%K A209349 nonn,tabl,walk
%O A209349 0,18
%A A209349 _Alois P. Heinz_, Mar 06 2012