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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.

%I A354127 #14 May 28 2022 12:48:08
%S A354127 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 A354127 900,110,6,0,51386,74086,43225,13020,2135,182,7,0,498026,793490,
%U A354127 524784,185920,37940,4452,280,8,0,5019720,8761906,6475959,2634912,642180,95508,8442,408,9,0
%N 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.
%H A354127 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 A354127 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 A354127 T(n, n-2) = A006331(n-1).
%e A354127 The triangle begins
%e A354127       1;
%e A354127       1,   0;
%e A354127       2,   2,   0;
%e A354127      12,  10,   3,   0;
%e A354127      82,  82,  28,   4,   0;
%e A354127     646, 738, 315,  60,   5,   0;
%e A354127     ...
%t A354127 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 A354127 Cf. A000007 (k = n), A000027 (k = n - 1), A000108, A001246 (row sums), A006331, A007318, A052553.
%K A354127 nonn,tabl
%O A354127 0,4
%A A354127 _Stefano Spezia_, May 18 2022