A354127 Triangle read by rows: T(n, k) is the number of graphs obtained by adding k pierced circles to a path graph P_n.

1, 1, 0, 2, 2, 0, 12, 10, 3, 0, 82, 82, 28, 4, 0, 646, 738, 315, 60, 5, 0, 5574, 7198, 3636, 900, 110, 6, 0, 51386, 74086, 43225, 13020, 2135, 182, 7, 0, 498026, 793490, 524784, 185920, 37940, 4452, 280, 8, 0, 5019720, 8761906, 6475959, 2634912, 642180, 95508, 8442, 408, 9, 0

Offset: 0

Author

Stefano Spezia, May 18 2022

Keywords

• nonn
• tabl

Examples

			The triangle begins
      1;
      1,   0;
      2,   2,   0;
     12,  10,   3,   0;
     82,  82,  28,   4,   0;
    646, 738, 315,  60,   5,   0;
    ...
		

Links

• Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.

Crossrefs

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

Programs

Mathematica

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}]]

Formula

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).

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