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%I #9 Aug 30 2019 21:46:36
%S 1,1,1,1,4,11,32,99,326,1123,4064,15291,59924,242945,1019584,4409233,
%T 19648674,89938705,422744384,2035739041,10039057524,50610247483,
%U 260704414816,1370387233859,7346982653702,40131663286851,223238920709024,1263531826402891,7273434344119460
%N Number of set partitions of {1, ..., n} whose blocks are all singletons and pairs, not including {1, n} or {i, i + 1} for any i.
%C Also the number of spanning subgraphs of the complement of an n-cycle, with no overlapping edges.
%H Andrew Howroyd, <a href="/A306419/b306419.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*A034807(n, k)*A000085(n-2*k) for n > 2. - _Andrew Howroyd_, Aug 30 2019
%e The a(1) = 1 through a(5) = 11 set partitions:
%e {{1}} {{1}{2}} {{1}{2}{3}} {{13}{24}} {{1}{24}{35}}
%e {{1}{24}{3}} {{13}{24}{5}}
%e {{13}{2}{4}} {{13}{25}{4}}
%e {{1}{2}{3}{4}} {{14}{2}{35}}
%e {{14}{25}{3}}
%e {{1}{2}{35}{4}}
%e {{1}{24}{3}{5}}
%e {{1}{25}{3}{4}}
%e {{13}{2}{4}{5}}
%e {{14}{2}{3}{5}}
%e {{1}{2}{3}{4}{5}}
%t stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
%t Table[Length[stableSets[Complement[Subsets[Range[n],{2}],Sort/@Partition[Range[n],2,1,1]],Intersection[#1,#2]!={}&]],{n,0,10}]
%o (PARI) \\ here b(n) is A000085(n)
%o b(n) = {sum(k=0, n\2, n!/((n-2*k)!*2^k*k!))}
%o a(n) = {if(n < 3, n >= 0, sum(k=0, n\2, (-1)^k*b(n-2*k)*n*(n-1-k)!/(k!*(n-2*k)!)))} \\ _Andrew Howroyd_, Aug 30 2019
%Y Cf. A000085, A000110, A000296, A001006, A001610, A003436 (no singletons), A034807, A170941 (linear case), A278990 (linear case with no singletons), A306417.
%K nonn
%O 0,5
%A _Gus Wiseman_, Feb 14 2019
%E Terms a(16) and beyond from _Andrew Howroyd_, Aug 30 2019
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