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A287376 Array read by antidiagonals: T(m,n) = number of independent vertex sets in the complete prism graph K_m X C_n.

%I A287376 #10 Jun 06 2017 11:54:35
%S A287376 1,3,1,4,7,1,7,13,13,1,11,35,34,21,1,18,81,121,73,31,1,29,199,391,325,
%T A287376 136,43,1,47,477,1300,1361,731,229,57,1,76,1155,4285,5781,3771,1447,
%U A287376 358,73,1,123,2785,14161,24473,19606,8881,2605,529,91,1
%N A287376 Array read by antidiagonals: T(m,n) = number of independent vertex sets in the complete prism graph K_m X C_n.
%C A287376 Equivalently, the number of 0..m words of length n with cyclically adjacent letters unequal with the exception that 0's may be adjacent.
%H A287376 Andrew Howroyd, <a href="/A287376/b287376.txt">Table of n, a(n) for n = 1..1275</a>
%F A287376 Row g.f.: ((m+1)-(m^2-2)*x-(2*m-1)*x^2)/(1-(m-1)*x-(m+1)*x^2-x^3).
%e A287376 Table starts:
%e A287376 ====================================================
%e A287376 m\n| 1  2   3    4     5      6       7        8
%e A287376 ---|------------------------------------------------
%e A287376 1  | 1  3   4    7    11     18      29       47 ...
%e A287376 2  | 1  7  13   35    81    199     477     1155 ...
%e A287376 3  | 1 13  34  121   391   1300    4285    14161 ...
%e A287376 4  | 1 21  73  325  1361   5781   24473   103685 ...
%e A287376 5  | 1 31 136  731  3771  19606  101781   528531 ...
%e A287376 6  | 1 43 229 1447  8881  54763  337429  2079367 ...
%e A287376 7  | 1 57 358 2605 18551 132504  946037  6754805 ...
%e A287376 8  | 1 73 529 4361 35361 287305 2333745 18957321 ...
%e A287376 ...
%t A287376 max = 10; row[m_] := ((m+1) - (m^2 - 2)*x - (2*m - 1)*x^2)/(1 - (m-1)*x - (m+1)*x^2 - x^3) + O[x]^(max+1) // CoefficientList[#, x]& // Rest;
%t A287376 T = Table[row[m], {m, 1, max}];
%t A287376 Table[T[[m-n+1, n]], {m, 1, max}, {n, m, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 06 2017 *)
%o A287376 (PARI)
%o A287376 RowGf(m,x)=((m+1)-(m^2-2)*x-(2*m-1)*x^2)/(1-(m-1)*x-(m+1)*x^2-x^3);
%o A287376 for (m=1,8,for(n=1,8,print1(Vec(RowGf(m,x)+O(x^(n+1)))[n+1], " "));print);
%Y A287376 Rows 2-7 are A051927, A051928, A051929, A051930, A051931, A051932.
%Y A287376 Cf. A135597 (K_m X P_n), A106512, A175243.
%K A287376 nonn,tabl
%O A287376 1,2
%A A287376 _Andrew Howroyd_, May 23 2017