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A323953 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.

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