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%I #8 Jan 19 2023 14:40:44
%S 1,2,1,5,2,1,12,6,2,1,27,12,7,2,1,58,23,14,8,2,1,121,44,23,16,9,2,1,
%T 248,82,38,26,18,10,2,1,503,149,65,38,29,20,11,2,1,1014,267,112,57,42,
%U 32,22,12,2,1,2037,475,189,90,57,46,35,24,13,2,1
%N Regular triangle read by rows where T(n, k) is the number of ways to split an n-cycle into singletons and connected subsequences of sizes > k.
%H Andrew Howroyd, <a href="/A323953/b323953.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F T(n,k) = 2 - n + Sum_{i=1..floor(n/k)} n*binomial(n-i*k+i-1, 2*i-1)/i for 1 <= k < n. - _Andrew Howroyd_, Jan 19 2023
%e Triangle begins:
%e 1
%e 2 1
%e 5 2 1
%e 12 6 2 1
%e 27 12 7 2 1
%e 58 23 14 8 2 1
%e 121 44 23 16 9 2 1
%e 248 82 38 26 18 10 2 1
%e 503 149 65 38 29 20 11 2 1
%e 1014 267 112 57 42 32 22 12 2 1
%e 2037 475 189 90 57 46 35 24 13 2 1
%e 4084 841 312 146 80 62 50 38 26 14 2 1
%e Row 4 counts the following connected partitions:
%e {{1234}} {{1234}} {{1234}} {{1}{2}{3}{4}}
%e {{1}{234}} {{1}{234}} {{1}{2}{3}{4}}
%e {{12}{34}} {{123}{4}}
%e {{123}{4}} {{124}{3}}
%e {{124}{3}} {{134}{2}}
%e {{134}{2}} {{1}{2}{3}{4}}
%e {{14}{23}}
%e {{1}{2}{34}}
%e {{1}{23}{4}}
%e {{12}{3}{4}}
%e {{14}{2}{3}}
%e {{1}{2}{3}{4}}
%t cyceds[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,Prepend[Range[i+k,n+i-1],i]}]];
%t spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t Table[Length[spsu[cyceds[n,k],Range[n]]],{n,10},{k,n}]
%o (PARI) T(n,k) = {1 + if(k<n, 1-n) + sum(i=1, n\k, n*binomial(n-i*k+i-1, 2*i-1)/i)} \\ _Andrew Howroyd_, Jan 19 2023
%Y First column is A000325. Second column is A323950.
%Y Cf. A001610, A001680, A005251, A066982, A306351, A323951, A323952, A323954.
%K nonn,tabl
%O 1,2
%A _Gus Wiseman_, Feb 10 2019
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