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A135597
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Square array read by antidiagonals: row m (m >= 1) satisfies b(0) = b(1) = 1; b(n) = m*b(n-1) + b(n-2):
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11
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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
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OFFSET
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1,6
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COMMENTS
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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
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LINKS
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FORMULA
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O.g.f. row m: (mx-x-1)/(x^2+mx-1). - R. J. Mathar, Apr 21 2008
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EXAMPLE
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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 ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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