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

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, 136, 43, 1, 47, 477, 1300, 1361, 731, 229, 57, 1, 76, 1155, 4285, 5781, 3771, 1447, 358, 73, 1, 123, 2785, 14161, 24473, 19606, 8881, 2605, 529, 91, 1

Offset

1,2

Comments

Equivalently, the number of 0..m words of length n with cyclically adjacent letters unequal with the exception that 0's may be adjacent.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Formula

Row g.f.: ((m+1)-(m^2-2)*x-(2*m-1)*x^2)/(1-(m-1)*x-(m+1)*x^2-x^3).

Example

Table starts:
====================================================
m\n| 1  2   3    4     5      6       7        8
---|------------------------------------------------
1  | 1  3   4    7    11     18      29       47 ...
2  | 1  7  13   35    81    199     477     1155 ...
3  | 1 13  34  121   391   1300    4285    14161 ...
4  | 1 21  73  325  1361   5781   24473   103685 ...
5  | 1 31 136  731  3771  19606  101781   528531 ...
6  | 1 43 229 1447  8881  54763  337429  2079367 ...
7  | 1 57 358 2605 18551 132504  946037  6754805 ...
8  | 1 73 529 4361 35361 287305 2333745 18957321 ...
...

Mathematica

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 = Table[row[m], {m, 1, max}];
Table[T[[m-n+1, n]], {m, 1, max}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Jun 06 2017 *)

Prog

(PARI)
RowGf(m, x)=((m+1)-(m^2-2)*x-(2*m-1)*x^2)/(1-(m-1)*x-(m+1)*x^2-x^3);
for (m=1, 8, for(n=1, 8, print1(Vec(RowGf(m, x)+O(x^(n+1)))[n+1], " ")); print);

Crossrefs

Rows 2-7 are A051927, A051928, A051929, A051930, A051931, A051932.

Cf. A135597 (K_m X P_n), A106512, A175243.

Sequence in context: A208339 A328463 A185722 * A209418 A193969 A169838

Adjacent sequences: A287373 A287374 A287375 * A287377 A287378 A287379

Keyword

nonn,tabl

Author

Andrew Howroyd, May 23 2017

Status

approved