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A209352 Number of initially rising meander words, where each letter of the cyclic 6-ary alphabet occurs n times.

%I A209352 #17 May 14 2020 05:35:25
%S A209352 1,1,16,484,17956,749956,33779344,1603842304,79171327876,
%T A209352 4026836863204,209730177700096,11135960392243600,600800844868633600,
%U A209352 32853035097265158400,1817225079550242841600,101519847275313821814784,5720749624907993103318916,324836041052683988251601956
%N A209352 Number of initially rising meander words, where each letter of the cyclic 6-ary alphabet occurs n times.
%C A209352 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 A209352 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 A209352 a(n) is also the number of (6*n-1)-step walks on 6-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 {1,6}.
%H A209352 Alois P. Heinz, <a href="/A209352/b209352.txt">Table of n, a(n) for n = 0..500</a>
%F A209352 a(n) = A197657(n-1)^2 for n>0, a(0) = 1.
%F A209352 a(n) ~ 3 * 2^(6*n - 4) / (Pi^2 * n^2). - _Vaclav Kotesovec_, May 14 2020
%e A209352 a(0) =  1: the empty word.
%e A209352 a(1) =  1 = |{abcdef}|.
%e A209352 a(2) = 16 = |{ababcdcdefef, abafedcbcdef, abafefedcbcd, abafefedcdcb, abcbafedcdef, abcbafefedcd, abcbcdedefaf, abcbcdefafed, abcdcbafedef, abcdcbafefed, abcdcdefefab, abcdedcbafef, abcdefabcdef, abcdefafedcb, abcdefedcbaf, abcdefefabcd}|.
%p A209352 g:= proc(m, n, k) local h;
%p A209352       h:= binomial(n-1, k);
%p A209352       h^m +`if`(m<2, 0, h* g(m-1, n, n-k-2))
%p A209352     end:
%p A209352 a:= n-> add(g(3, n, k), k=0..n)^2:
%p A209352 seq(a(n), n=0..30);
%t A209352 g[m_, n_, k_] := g[m, n, k] = With[{h = Binomial[n - 1, k]}, h^m + If[m < 2, 0, h g[m - 1, n, n - k - 2]]];
%t A209352 a[n_] := Sum[g[3, n, k], {k, 0, n}]^2;
%t A209352 a /@ Range[0, 30] (* _Jean-François Alcover_, May 14 2020, after Maple *)
%Y A209352 Column k=6 of A209349.
%Y A209352 Cf. A197657.
%K A209352 nonn,walk
%O A209352 0,3
%A A209352 _Alois P. Heinz_, Mar 06 2012