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A290724
Triangle read by rows: T(n,k) = number of arrangements of k non-attacking rooks on an n X n right triangular board with every square controlled by at least one rook.
2
1, 1, 1, 0, 4, 1, 0, 2, 11, 1, 0, 0, 18, 26, 1, 0, 0, 6, 100, 57, 1, 0, 0, 0, 96, 444, 120, 1, 0, 0, 0, 24, 900, 1734, 247, 1, 0, 0, 0, 0, 600, 6480, 6246, 502, 1, 0, 0, 0, 0, 120, 8520, 39762, 21320, 1013, 1, 0, 0, 0, 0, 0, 4320, 90600, 219312, 70128, 2036, 1
OFFSET
1,5
COMMENTS
See A146304 for algorithm and PARI code to produce this sequence.
Equivalently, the number of maximal independent vertex sets in the n-triangular honeycomb bishop graph with k vertices. A bishop can move along two axes in the triangular honeycomb grid.
LINKS
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
EXAMPLE
Triangle begins:
1;
1, 1;
0, 4, 1;
0, 2, 11, 1;
0, 0, 18, 26, 1;
0, 0, 6, 100, 57, 1;
0, 0, 0, 96, 444, 120, 1;
0, 0, 0, 24, 900, 1734, 247, 1;
0, 0, 0, 0, 600, 6480, 6246, 502, 1;
0, 0, 0, 0, 120, 8520, 39762, 21320, 1013, 1;
...
MATHEMATICA
CoefficientList[Table[Sum[k! StirlingS2[m, k] StirlingS2[n + 1 - m, k + 1] x^(n - k), {m, 0, n}, {k, 0, Min[m, n - m]}], {n, 20}]/x, x] // Flatten (* Eric W. Weisstein, Feb 01 2024 *)
CROSSREFS
Row sums are A290615.
Sequence in context: A206799 A084119 A166073 * A283879 A216178 A122899
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Aug 09 2017
STATUS
approved