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A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.

%I A216078 #70 Feb 16 2025 08:33:18
%S A216078 1,1,3,7,27,87,409,1657,9089,43833,272947,1515903,10515147,65766991,
%T A216078 501178937,3473600465,28773452321,218310229201,1949230218691,
%U A216078 16035686850327,153281759047387,1356791248984295,13806215066685433,130660110400259849,1408621900803060705
%N A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.
%C A216078 Number of vertex covers and independent vertex sets of the n-1 X n-1 white bishops graph. Equivalently, the number of ways to place any number of non-attacking bishops on the white squares of an n-1 X n-1 board. - _Andrew Howroyd_, May 08 2017
%C A216078 Number of pairs of partitions (A<=B) of [n-1] such that the nontrivial blocks of A are of type {k,n-1-k} if n is even, and of type {k,n-k} if n is odd. - _Francesca Aicardi_, May 28 2022
%H A216078 R. H. Hardin, <a href="/A216078/b216078.txt">Table of n, a(n) for n = 1..210</a>
%H A216078 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H A216078 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H A216078 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WhiteBishopGraph.html">White Bishop Graph</a>
%F A216078 a(n) = Sum_{k=0..m} binomial(m, k)*Bell(m+k+e), with m = floor((n-1)/2), e = (n+1) mod 2 and where Bell(n) is the Bell exponential number A000110(n). - _Francesca Aicardi_, May 28 2022
%F A216078 From _Vaclav Kotesovec_, Jul 29 2022: (Start)
%F A216078 a(2*k) = A020556(k).
%F A216078 a(2*k+1) = A094577(k). (End)
%e A216078 Some solutions for n = 5:
%e A216078   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0
%e A216078   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x
%e A216078   x 2   x 0   x 0   x 2   x 0   x 1   x 1   x 2   x 2   x 1
%e A216078   0 x   2 x   1 x   3 x   1 x   0 x   2 x   3 x   0 x   0 x
%e A216078   x 3   x 1   x 2   x 2   x 0   x 1   x 1   x 1   x 2   x 0
%e A216078 There are 4 white squares on a 3 X 3 board. There is 1 way to place no non-attacking bishops, 4 ways to place 1 and 2 ways to place 2 so a(4) = 1 + 4 + 2 = 7. - _Andrew Howroyd_, Jun 06 2017
%p A216078 a:= n-> (m-> add(binomial(m, k)*combinat[bell](m+k+e)
%p A216078            , k=0..m))(iquo(n-1, 2, 'e')):
%p A216078 seq(a(n), n=1..26);  # _Alois P. Heinz_, Oct 03 2022
%t A216078 a[n_] := Module[{m, e}, {m, e} = QuotientRemainder[n - 1, 2];
%t A216078    Sum[Binomial[m, k]*BellB[m + k + e], {k, 0, m}]];
%t A216078 Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Jul 25 2022, after _Francesca Aicardi_ *)
%Y A216078 Column 2 of A216084.
%Y A216078 Row sums of A274106(n-1).
%Y A216078 Cf. A020556, A094577, A216332, A201862, A286423.
%K A216078 nonn
%O A216078 1,3
%A A216078 _R. H. Hardin_, Sep 01 2012