cp's OEIS Frontend

A135597 Square array read by antidiagonals: row m (m >= 1) satisfies b(0) = b(1) = 1; b(n) = m*b(n-1) + b(n-2).

%I A135597 #26 Jun 01 2025 21:27:41
%S A135597 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 A135597 31,89,142,99,21,1,1,8,43,161,377,469,239,34,1,1,9,57,265,836,1597,
%U A135597 1549,577,55,1,1,10,73,407,1633,4341,6765,5116,1393,89,1,1,11,91,593,2906
%N A135597 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 A135597 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 A135597 Andrew Howroyd, <a href="/A135597/b135597.txt">Table of n, a(n) for n = 1..1275</a>
%F A135597 O.g.f. row m: (mx-x-1)/(x^2+mx-1). - _R. J. Mathar_, Apr 21 2008
%e A135597 Array begins:
%e A135597 ========================================================
%e A135597 m\n| 0 1 2  3   4    5     6      7       8        9
%e A135597 ---|----------------------------------------------------
%e A135597 1  | 1 1 2  3   5    8    13     21      34       55 ...
%e A135597 2  | 1 1 3  7  17   41    99    239     577     1393 ...
%e A135597 3  | 1 1 4 13  43  142   469   1549    5116    16897 ...
%e A135597 4  | 1 1 5 21  89  377  1597   6765   28657   121393 ...
%e A135597 5  | 1 1 6 31 161  836  4341  22541  117046   607771 ...
%e A135597 6  | 1 1 7 43 265 1633 10063  62011  382129  2354785 ...
%e A135597 7  | 1 1 8 57 407 2906 20749 148149 1057792  7552693 ...
%e A135597 8  | 1 1 9 73 593 4817 39129 317849 2581921 20973217 ...
%e A135597 ...
%p A135597 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 A135597 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 A135597 Cf. A121875, A000045, A287376.
%Y A135597 Rows 2-11 are A001333, A003688, A015448, A015449, A015451, A015453-A015457.
%K A135597 nonn,tabl,easy
%O A135597 1,6
%A A135597 _N. J. A. Sloane_, Mar 02 2008
%E A135597 More terms from _R. J. Mathar_, Apr 21 2008