A324428 - OEIS

%I #26 Apr 27 2020 06:31:59

%S 1,3,1,15,4,1,105,31,7,1,945,293,68,11,1,10395,3326,837,159,18,1,

%T 135135,44189,11863,2488,381,29,1,2027025,673471,189503,43169,7601,

%U 879,47,1,34459425,11588884,3377341,822113,160784,23559,2049,76,1,654729075,222304897,66564396,17066007,3621067,607897,72989,4788,123,1

%N Number T(n,k) of labeled cyclic chord diagrams with n chords such that every chord has length at least k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%C T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with 1 <= k <= n. T(n,0) = A001147(n), T(0,k) = 1, T(n,k) = 0 for k > n > 0.

%H Alois P. Heinz, <a href="/A324428/b324428.txt">Rows n = 1..19, flattened</a>

%F T(n,k) = Sum_{j=k..n} A324429(n,j).

%e Triangle T(n,k) begins:

%e         1;

%e         3,      1;

%e        15,      4,      1;

%e       105,     31,      7,     1;

%e       945,    293,     68,    11,    1;

%e     10395,   3326,    837,   159,   18,   1;

%e    135135,  44189,  11863,  2488,  381,  29,  1;

%e   2027025, 673471, 189503, 43169, 7601, 879, 47, 1;

%e   ...

%p b:= proc(n, f, m, l, j) option remember; (k-> `if`(n<add(i, i=f)+m+

%p       add(i, i=l), 0, `if`(n=0, 1, add(`if`(f[i]=0, 0, b(n-1,

%p       subsop(i=0, f), m+l[1], [subsop(1=[][], l)[], 0], max(0, j-1))),

%p       i=max(1, j+1)..min(k, n-1))+`if`(m=0, 0, m*b(n-1, f, m-1+l[1],

%p       [subsop(1=[][], l)[], 0], max(0, j-1)))+b(n-1, f, m+l[1],

%p       [subsop(1=[][], l)[], 1], max(0, j-1)))))(nops(l))

%p     end:

%p T:= (n, k)-> `if`(n=0 or k<2, doublefactorial(2*n-1),

%p               b(2*n-k+1, [1$k-1], 0, [0$k-1], k-1)):

%p seq(seq(T(n, k), k=1..n), n=1..10);

%t b[n_, f_List, m_, l_List, j_] := b[n, f, m, l, j] = Function[k, If[n < Total[f] + m + Total[l], 0, If[n == 0, 1, Sum[If[f[[i]] == 0, 0, b[n - 1, ReplacePart[f, i -> 0], m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]],{i, Max[1, j + 1], Min[k, n - 1]}] + If[m == 0, 0, m*b[n - 1, f, m - 1 + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]] + b[n - 1, f, m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 1], Max[0, j - 1]]]]][Length[l]];

%t T[n_, k_] := If[n == 0 || k < 2, 2^(n-1) Pochhammer[3/2, n-1], b[2n-k+1, Table[1, {k-1}], 0, Table[0, {k-1}], k-1]];

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 27 2020, after _Alois P. Heinz_ *)

%Y Columns k=1-10 give: A001147, A003436, A306386, A324430, A324431, A324432, A324433, A324434, A324435, A324436.

%Y T(n,n-1) gives A000204.

%Y Cf. A293157, A293881, A324429.

%K nonn,tabl

%O 1,2

%A _Alois P. Heinz_, Feb 27 2019