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A276562 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 1 or less.

%I A276562 #30 Jun 06 2017 11:54:26
%S A276562 1,1,2,1,4,3,1,8,7,4,1,16,15,10,5,1,32,35,22,13,6,1,64,83,54,29,16,7,
%T A276562 1,128,199,134,73,36,19,8,1,256,479,340,185,92,43,22,9,1,512,1155,872,
%U A276562 481,236,111,50,25,10,1,1024,2787,2254,1265,622,287,130,57,28,11
%N A276562 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 1 or less.
%C A276562 All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions.
%H A276562 Andrew Howroyd, <a href="/A276562/b276562.txt">Table of n, a(n) for n = 1..1275</a>
%H A276562 Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, <a href="http://arxiv.org/abs/0809.0551">Smooth words and Chebyshev polynomials</a>, arXiv:0809.0551v1 [math.CO], 2008.
%F A276562 T(m, n) = Sum_{j=1..m} (1 + 2*cos(j*pi/(m+1)))^n. - _Andrew Howroyd_, Apr 15 2017
%e A276562 Array starts:
%e A276562    1  1  1   1   1    1    1    1     1     1 ...
%e A276562    2  4  8  16  32   64  128  256   512  1024 ...
%e A276562    3  7 15  35  83  199  479 1155  2787  6727 ...
%e A276562    4 10 22  54 134  340  872 2254  5854 15250 ...
%e A276562    5 13 29  73 185  481 1265 3361  8993 24193 ...
%e A276562    6 16 36  92 236  622 1658 4468 12132 33146 ...
%e A276562    7 19 43 111 287  763 2051 5575 15271 42099 ...
%e A276562    8 22 50 130 338  904 2444 6682 18410 51052 ...
%e A276562    9 25 57 149 389 1045 2837 7789 21549 60005 ...
%e A276562   10 28 64 168 440 1186 3230 8896 24688 68958 ...
%t A276562 T[m_, n_] := Sum[(1 + 2*Cos[j*Pi/(m+1)])^n, {j, 1, m}] // FullSimplify;
%t A276562 Table[T[m-n+1, n], {m, 1, 11}, {n, m, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 06 2017 *)
%o A276562 (PARI) \\ from Knopfmacher et al.
%o A276562 ChebyshevU(n,x) = sum(i=0, n/2, 2*poltchebi(n-2*i,x)) + (n%2-1);
%o A276562 RowGf(k,x) = 1 + (k*x*(1+3*x) - 2*(k+1)*x*subst(ChebyshevU(k-1,z)/ChebyshevU(k,z),z,(1-x)/(2*x)))/((1+x)*(1-3*x));
%o A276562 a(m,n)=Vec(RowGf(m,x)+O(x^(n+1)))[n+1];
%o A276562 for(m=1, 10, print(RowGf(m,x)));
%o A276562 for(m=1, 10, for(n=1, 9, print1( a(m,n), ", ") ); print(); );
%Y A276562 Rows 3-32 are A124696-A124719, A124726, A124783, A124784, A124799, A124803, A124804.
%Y A276562 Cf. A188866, A220062, A285280, A285281, A208777, A208721.
%K A276562 nonn,tabl
%O A276562 1,3
%A A276562 _Andrew Howroyd_, Apr 15 2017