%I #24 May 25 2017 11:56:42

%S 1,1,1,1,1,2,1,1,3,3,1,1,4,7,5,1,1,5,13,17,8,1,1,6,21,43,41,13,1,1,7,

%T 31,89,142,99,21,1,1,8,43,161,377,469,239,34,1,1,9,57,265,836,1597,

%U 1549,577,55,1,1,10,73,407,1633,4341,6765,5116,1393,89,1,1,11,91,593,2906

%N Square array read by antidiagonals: row m (m >= 1) satisfies b(0) = b(1) = 1; b(n) = m*b(n-1) + b(n-2):

%C For n > 1, the number of independent vertex sets in the graph K_m X P_{n-1}. For example, in K_3 X P_1 there are 4 independent vertex sets. - _Andrew Howroyd_, May 23 2017

%H Andrew Howroyd, <a href="/A135597/b135597.txt">Table of n, a(n) for n = 1..1275</a>

%F O.g.f. row m: (mx-x-1)/(x^2+mx-1). - _R. J. Mathar_, Apr 21 2008

%e Array begins:

%e ========================================================

%e m\n| 0 1 2 3 4 5 6 7 8 9

%e ---|----------------------------------------------------

%e 1 | 1 1 2 3 5 8 13 21 34 55 ...

%e 2 | 1 1 3 7 17 41 99 239 577 1393 ...

%e 3 | 1 1 4 13 43 142 469 1549 5116 16897 ...

%e 4 | 1 1 5 21 89 377 1597 6765 28657 121393 ...

%e 5 | 1 1 6 31 161 836 4341 22541 117046 607771 ...

%e 6 | 1 1 7 43 265 1633 10063 62011 382129 2354785 ...

%e 7 | 1 1 8 57 407 2906 20749 148149 1057792 7552693 ...

%e 8 | 1 1 9 73 593 4817 39129 317849 2581921 20973217 ...

%e ...

%p A135597 := proc(m,c) coeftayl( (m*x-x-1)/(x^2+m*x-1),x=0,c) ; end: for d from 1 to 15 do for c from 0 to d-1 do printf("%d,",A135597(d-c,c)) ; od: od: # _R. J. Mathar_, Apr 21 2008

%t a[_, 0] = a[_, 1] = 1; a[m_, n_] := m*a[m, n-1] + a[m, n-2]; Table[a[m-n+1, n], {m, 0, 11}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Jan 20 2014 *)

%Y Cf. A121875, A000045, A287376.

%Y Rows 2-11 are A001333, A003688, A015448, A015449, A015451, A015453-A015457.

%K nonn,tabl,easy

%O 1,6

%A _N. J. A. Sloane_, Mar 02 2008

%E More terms from _R. J. Mathar_, Apr 21 2008