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).

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, 31, 89, 142, 99, 21, 1, 1, 8, 43, 161, 377, 469, 239, 34, 1, 1, 9, 57, 265, 836, 1597, 1549, 577, 55, 1, 1, 10, 73, 407, 1633, 4341, 6765, 5116, 1393, 89, 1, 1, 11, 91, 593, 2906

Offset: 1

Author

N. J. A. Sloane, Mar 02 2008

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			Array begins:
========================================================
m\n| 0 1 2  3   4    5     6      7       8        9
---|----------------------------------------------------
1  | 1 1 2  3   5    8    13     21      34       55 ...
2  | 1 1 3  7  17   41    99    239     577     1393 ...
3  | 1 1 4 13  43  142   469   1549    5116    16897 ...
4  | 1 1 5 21  89  377  1597   6765   28657   121393 ...
5  | 1 1 6 31 161  836  4341  22541  117046   607771 ...
6  | 1 1 7 43 265 1633 10063  62011  382129  2354785 ...
7  | 1 1 8 57 407 2906 20749 148149 1057792  7552693 ...
8  | 1 1 9 73 593 4817 39129 317849 2581921 20973217 ...
...
		

Links

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

Crossrefs

Cf. A121875, A000045, A287376.

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

Programs

Maple

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

Mathematica

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 *)

Formula

O.g.f. row m: (mx-x-1)/(x^2+mx-1). - R. J. Mathar, Apr 21 2008

Extensions

More terms from R. J. Mathar, Apr 21 2008