A354127 - OEIS

%I #14 May 28 2022 12:48:08

%S 1,1,0,2,2,0,12,10,3,0,82,82,28,4,0,646,738,315,60,5,0,5574,7198,3636,

%T 900,110,6,0,51386,74086,43225,13020,2135,182,7,0,498026,793490,

%U 524784,185920,37940,4452,280,8,0,5019720,8761906,6475959,2634912,642180,95508,8442,408,9,0

%N Triangle read by rows: T(n, k) is the number of graphs obtained by adding k pierced circles to a path graph P_n.

%H Nicholas Owad and Anastasiia Tsvietkova, <a href="https://arxiv.org/abs/2205.03451">Random meander model for links</a>, arXiv:2205.03451 [math.GT], 2022.

%F T(n, k) = Sum_{m=k..n} (-1)^(m+k)*binomial(m, k)*O(m, n), with O(k, s) = binomial(2*s-k-1, k)*C(s-k)^2 (see Lemma 3.3 at page 7 in Owad and Tsvietkova).

%F T(n, n-2) = A006331(n-1).

%e The triangle begins

%e       1;

%e       1,   0;

%e       2,   2,   0;

%e      12,  10,   3,   0;

%e      82,  82,  28,   4,   0;

%e     646, 738, 315,  60,   5,   0;

%e     ...

%t bigO[k_,s_]:=Binomial[2s-k-1,k]CatalanNumber[s-k]^2; T[n_,k_]:=Sum[(-1)^(m+k)Binomial[m,k]bigO[m,n],{m,k,n}];Flatten[Table[T[n,k],{n,0,9},{k,0,n}]]

%Y Cf. A000007 (k = n), A000027 (k = n - 1), A000108, A001246 (row sums), A006331, A007318, A052553.

%K nonn,tabl

%O 0,4

%A _Stefano Spezia_, May 18 2022