A290724 - OEIS

%I #12 Feb 01 2024 08:21:44

%S 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,

%T 0,0,24,900,1734,247,1,0,0,0,0,600,6480,6246,502,1,0,0,0,0,120,8520,

%U 39762,21320,1013,1,0,0,0,0,0,4320,90600,219312,70128,2036,1

%N 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.

%C See A146304 for algorithm and PARI code to produce this sequence.

%C 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.

%H Andrew Howroyd, <a href="/A290724/b290724.txt">Table of n, a(n) for n = 1..1275</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIndependentVertexSet.html">Maximal Independent Vertex Set</a>

%e Triangle begins:

%e 1;

%e 1, 1;

%e 0, 4,  1;

%e 0, 2, 11,  1;

%e 0, 0, 18,  26,  1;

%e 0, 0,  6, 100,  57,    1;

%e 0, 0,  0,  96, 444,  120,     1;

%e 0, 0,  0,  24, 900, 1734,   247,     1;

%e 0, 0,  0,  0,  600, 6480,  6246,   502,    1;

%e 0, 0,  0,  0,  120, 8520, 39762, 21320, 1013, 1;

%e ...

%t 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 *)

%Y Row sums are A290615.

%Y Cf. A259691, A259697.

%K nonn,tabl

%O 1,5

%A _Andrew Howroyd_, Aug 09 2017